P. 353 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35201-35300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bnj1493 35201*	Technical lemma for bnj60 35204. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    ⇒   ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
Theorem	bnj1497 35202*	Technical lemma for bnj60 35204. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    ⇒   ⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔
Theorem	bnj1498 35203*	Technical lemma for bnj60 35204. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    ⇒   ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
21.4.5  Well-founded recursion, part 1 of 3
Theorem	bnj60 35204*	Well-founded recursion, part 1 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    ⇒   ⊢ (𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴)
Theorem	bnj1514 35205*	Technical lemma for bnj1500 35210. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    ⇒   ⊢ (𝑓 ∈ 𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓‘𝑥) = (𝐺‘𝑌))
Theorem	bnj1518 35206*	Technical lemma for bnj1500 35210. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓))    ⇒   ⊢ (𝜓 → ∀𝑑𝜓)
Theorem	bnj1519 35207*	Technical lemma for bnj1500 35210. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    ⇒   ⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Theorem	bnj1520 35208*	Technical lemma for bnj1500 35210. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    ⇒   ⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Theorem	bnj1501 35209*	Technical lemma for bnj1500 35210. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑))    ⇒   ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
21.4.6  Well-founded recursion, part 2 of 3
Theorem	bnj1500 35210*	Well-founded recursion, part 2 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    ⇒   ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Theorem	bnj1525 35211*	Technical lemma for bnj1522 35214. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝐹 ≠ 𝐻))    ⇒   ⊢ (𝜓 → ∀𝑥𝜓)
Theorem	bnj1529 35212*	Technical lemma for bnj1522 35214. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜒 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))    &   ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹)    ⇒   ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))
Theorem	bnj1523 35213*	Technical lemma for bnj1522 35214. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    &   ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))    &   ⊢ (𝜓 ↔ (𝜑 ∧ 𝐹 ≠ 𝐻))    &   ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ≠ (𝐻‘𝑥)))    &   ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐻‘𝑥)}    &   ⊢ (𝜃 ↔ (𝜒 ∧ 𝑦 ∈ 𝐷 ∧ ∀𝑧 ∈ 𝐷 ¬ 𝑧𝑅𝑦))    ⇒   ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → 𝐹 = 𝐻)
21.4.7  Well-founded recursion, part 3 of 3
Theorem	bnj1522 35214*	Well-founded recursion, part 3 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}    &   ⊢ 𝐹 = ∪ 𝐶    ⇒   ⊢ ((𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → 𝐹 = 𝐻)
21.5  Mathbox for BTernaryTau
21.5.1  First-order logic
21.5.1.1  Auxiliary axiom schemes
Theorem	nfan1c 35215	Variant of nfan 1901 and commuted form of nfan1 2208. (Contributed by BTernaryTau, 31-Jul-2025.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ Ⅎ𝑥(𝜓 ∧ 𝜑)
Theorem	cbvex1v 35216*	Rule used to change bound variables, using implicit substitution. (Contributed by BTernaryTau, 31-Jul-2025.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑦𝜒))
Theorem	dvelimalcased 35217*	Eliminate a disjoint variable condition from a universally quantified statement using cases. (Contributed by BTernaryTau, 31-Jul-2025.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜃)    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑥 → (𝜓 → 𝜃)))    &   ⊢ ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (𝜒 → 𝜃))    &   ⊢ (𝜑 → ∀𝑧𝜓)    &   ⊢ (𝜑 → ∀𝑥𝜒)    ⇒   ⊢ (𝜑 → ∀𝑥𝜃)
Theorem	dvelimalcasei 35218*	Eliminate a disjoint variable condition from a universally quantified statement using cases. Inference form of dvelimalcased 35217. (Contributed by BTernaryTau, 31-Jul-2025.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜒)    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝜑 → 𝜒)))    &   ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜓 → 𝜒))    &   ⊢ ∀𝑧𝜑    &   ⊢ ∀𝑥𝜓    ⇒   ⊢ ∀𝑥𝜒
Theorem	dvelimexcased 35219*	Eliminate a disjoint variable condition from an existentially quantified statement using cases. (Contributed by BTernaryTau, 31-Jul-2025.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜃)    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑥 → (𝜓 → 𝜃)))    &   ⊢ ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (𝜒 → 𝜃))    &   ⊢ (𝜑 → ∃𝑧𝜓)    &   ⊢ (𝜑 → ∃𝑥𝜒)    ⇒   ⊢ (𝜑 → ∃𝑥𝜃)
Theorem	dvelimexcasei 35220*	Eliminate a disjoint variable condition from an existentially quantified statement using cases. Inference form of dvelimexcased 35219. See axnulg 35251 for an example of its use. (Contributed by BTernaryTau, 31-Jul-2025.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜒)    &   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝜑 → 𝜒)))    &   ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜓 → 𝜒))    &   ⊢ ∃𝑧𝜑    &   ⊢ ∃𝑥𝜓    ⇒   ⊢ ∃𝑥𝜒
21.5.2  ZF set theory
Theorem	exdifsn 35221	There exists an element in a class excluding a singleton if and only if there exists an element in the original class not equal to the singleton element. (Contributed by BTernaryTau, 15-Sep-2023.)
⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 𝐵)
Theorem	srcmpltd 35222	If a statement is true for every element of a class and for every element of its complement relative to a second class, then it is true for every element in the second class. (Contributed by BTernaryTau, 27-Sep-2023.)
⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝜓))    &   ⊢ (𝜑 → (𝐶 ∈ (𝐵 ∖ 𝐴) → 𝜓))    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐵 → 𝜓))
Theorem	prsrcmpltd 35223	If a statement is true for all pairs of elements of a class, all pairs of elements of its complement relative to a second class, and all pairs with one element in each, then it is true for all pairs of elements of the second class. (Contributed by BTernaryTau, 27-Sep-2023.)
⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → 𝜓))    &   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ (𝐵 ∖ 𝐴)) → 𝜓))    &   ⊢ (𝜑 → ((𝐶 ∈ (𝐵 ∖ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝜓))    &   ⊢ (𝜑 → ((𝐶 ∈ (𝐵 ∖ 𝐴) ∧ 𝐷 ∈ (𝐵 ∖ 𝐴)) → 𝜓))    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → 𝜓))
Theorem	axnulALT2 35224*	Alternate proof of axnul 5240, proved from propositional calculus, ax-gen 1797, ax-4 1811, ax-6 1969, and ax-rep 5212. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BTernaryTau, 27-Mar-2026.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	axsepg2 35225*	A generalization of ax-sep 5231 in which 𝑦 and 𝑧 need not be distinct. See also axsepg 5232 which instead allows 𝑧 to occur in 𝜑. Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by BTernaryTau, 3-Aug-2025.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	axsepg2ALT 35226*	Alternate proof of axsepg2 35225, derived directly from ax-sep 5231 with no additional set theory axioms. (Contributed by BTernaryTau, 3-Aug-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	dff15 35227*	A one-to-one function in terms of different arguments never having the same function value. (Contributed by BTernaryTau, 24-Oct-2023.)
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)))
Theorem	f1resveqaeq 35228	If a function restricted to a class is one-to-one, then for any two elements of the class, the values of the function at those elements are equal only if the two elements are the same element. (Contributed by BTernaryTau, 27-Sep-2023.)
⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))
Theorem	f1resrcmplf1dlem 35229	Lemma for f1resrcmplf1d 35230. (Contributed by BTernaryTau, 27-Sep-2023.)
⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐷 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅)    ⇒   ⊢ (𝜑 → ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐷) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
Theorem	f1resrcmplf1d 35230	If a function's restriction to a subclass of its domain and its restriction to the relative complement of that subclass are both one-to-one, and if the ranges of those two restrictions are disjoint, then the function is itself one-to-one. (Contributed by BTernaryTau, 28-Sep-2023.)
⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)    &   ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ 𝐶)):(𝐴 ∖ 𝐶)–1-1→𝐵)    &   ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ∅)    ⇒   ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
Theorem	funen1cnv 35231	If a function is equinumerous to ordinal 1, then its converse is also a function. (Contributed by BTernaryTau, 8-Oct-2023.)
⊢ ((Fun 𝐹 ∧ 𝐹 ≈ 1o) → Fun ◡𝐹)
Theorem	xoromon 35232	ω is either an ordinal set or the proper class of all ordinal sets, but not both. This is a stronger version of omon 7829. (Contributed by BTernaryTau, 25-Jan-2026.)
⊢ (ω ∈ On ⊻ ω = On)
Theorem	fissorduni 35233	The union (supremum) of a finite set of ordinals less than a nonzero ordinal class is an element of that ordinal class. (Contributed by BTernaryTau, 15-Jan-2026.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ (Ord 𝐵 ∧ 𝐵 ≠ ∅)) → ∪ 𝐴 ∈ 𝐵)
Theorem	fnrelpredd 35234*	A function that preserves a relation also preserves predecessors. (Contributed by BTernaryTau, 16-Jul-2024.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐴)    ⇒   ⊢ (𝜑 → Pred(𝑆, (𝐹 “ 𝐶), (𝐹‘𝐷)) = (𝐹 “ Pred(𝑅, 𝐶, 𝐷)))
Theorem	cardpred 35235	The cardinality function preserves predecessors. (Contributed by BTernaryTau, 18-Jul-2024.)
⊢ ((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝐵)) = (card “ Pred( ≺ , 𝐴, 𝐵)))
Theorem	nummin 35236*	Every nonempty class of numerable sets has a minimal element. (Contributed by BTernaryTau, 18-Jul-2024.)
⊢ ((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅)
Theorem	r11 35237	Value of the cumulative hierarchy of sets function at 1o. (Contributed by BTernaryTau, 24-Jan-2026.)
⊢ (𝑅1‘1o) = 1o
Theorem	r12 35238	Value of the cumulative hierarchy of sets function at 2o. (Contributed by BTernaryTau, 25-Jan-2026.)
⊢ (𝑅1‘2o) = 2o
Theorem	r1wf 35239	Each stage in the cumulative hierarchy is well-founded. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On)
Theorem	elwf 35240	An element of a well-founded set is well-founded. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ∪ (𝑅1 “ On))
Theorem	r1elcl 35241	Each set of the cumulative hierarchy is closed under membership. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ (𝑅1‘𝐵))
Theorem	rankval2b 35242*	Value of an alternate definition of the rank function. Definition of [BellMachover] p. 478. This variant of rankval2 9742 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅1. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (𝑅1‘𝑥)})
Theorem	rankval4b 35243*	The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204. This variant of rankval4 9791 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅1. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
Theorem	rankfilimbi 35244*	If all elements in a finite well-founded set have a rank less than a limit ordinal, then the rank of that set is also less than the limit ordinal. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)
Theorem	rankfilimb 35245*	The rank of a finite well-founded set is less than a limit ordinal iff the ranks of all of its elements are less than that limit ordinal. (Contributed by BTernaryTau, 22-Jan-2026.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵))
Theorem	r1filimi 35246*	If all elements in a finite set appear in the cumulative hierarchy prior to a limit ordinal, then that set also appears in the cumulative hierarchy prior to the limit ordinal. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ 𝐵))
Theorem	r1filim 35247*	A finite set appears in the cumulative hierarchy prior to a limit ordinal iff all of its elements appear in the cumulative hierarchy prior to that limit ordinal. (Contributed by BTernaryTau, 22-Jan-2026.)
⊢ ((𝐴 ∈ Fin ∧ Lim 𝐵) → (𝐴 ∈ ∪ (𝑅1 “ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵)))
Theorem	r1omfi 35248	Hereditarily finite sets are finite sets. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ ∪ (𝑅1 “ ω) ⊆ Fin
Theorem	r1omhf 35249*	A set is hereditarily finite iff it is finite and all of its elements are hereditarily finite. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ ω) ↔ (𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ ω)))
Theorem	r1ssel 35250	A set is a subset of the value of the cumulative hierarchy of sets function iff it is an element of the value at the successor. (Contributed by BTernaryTau, 15-Jan-2026.)
⊢ (𝐵 ∈ On → (𝐴 ⊆ (𝑅1‘𝐵) ↔ 𝐴 ∈ (𝑅1‘suc 𝐵)))
Theorem	axnulg 35251	A generalization of ax-nul 5241 in which 𝑥 and 𝑦 need not be distinct. Note that it is possible to use axc7e 2323 to derive elirrv 9512 from this theorem, which justifies the dependency on ax-reg 9507. Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by BTernaryTau, 3-Aug-2025.) (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	axnulALT3 35252*	Alternate proof of axnul 5240, proved from propositional calculus, ax-gen 1797, ax-4 1811, ax-5 1912, and ax-inf2 9562. (Contributed by BTernaryTau, 22-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
Theorem	axprALT2 35253*	Alternate proof of axpr 5369, proved from predicate calculus, ax-rep 5212, and ax-inf2 9562. (Contributed by BTernaryTau, 26-Mar-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
Theorem	r1omfv 35254	Value of the cumulative hierarchy of sets function at ω. (Contributed by BTernaryTau, 25-Jan-2026.)
⊢ (𝑅1‘ω) = ∪ (𝑅1 “ ω)
Theorem	trssfir1om 35255	If every element in a transitive class is finite, then every element is also hereditarily finite. (Contributed by BTernaryTau, 24-Jan-2026.)
⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ ∪ (𝑅1 “ ω))
Theorem	r1omhfb 35256*	The class of all hereditarily finite sets is the only class with the property that all sets are members of it iff they are finite and all of their elements are members of it. (Contributed by BTernaryTau, 24-Jan-2026.)
⊢ (𝐻 = ∪ (𝑅1 “ ω) ↔ ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
21.5.2.1  Finitism
Theorem	prcinf 35257*	Any proper class is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. This proof holds regardless of whether the Axiom of Infinity is accepted or negated. (Contributed by BTernaryTau, 22-Jun-2025.)
⊢ (¬ 𝐴 ∈ V → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))
Theorem	fineqvrep 35258*	If all sets are finite, then the Axiom of Replacement becomes redundant. (Contributed by BTernaryTau, 12-Sep-2024.)
⊢ (Fin = V → (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
Theorem	fineqvpow 35259*	If all sets are finite, then the Axiom of Power Sets becomes redundant. (Contributed by BTernaryTau, 12-Sep-2024.)
⊢ (Fin = V → ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
Theorem	fineqvac 35260	If all sets are finite, then the Axiom of Choice becomes redundant. For a shorter proof using ax-rep 5212 and ax-pow 5307, see fineqvacALT 35261. (Contributed by BTernaryTau, 21-Sep-2024.)
⊢ (Fin = V → CHOICE)
Theorem	fineqvacALT 35261	Shorter proof of fineqvac 35260 using ax-rep 5212 and ax-pow 5307. (Contributed by BTernaryTau, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (Fin = V → CHOICE)
Theorem	fineqvomon 35262	If all sets are finite, then the class of all natural numbers equals the proper class of all ordinal numbers. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (Fin = V → ω = On)
Theorem	fineqvomonb 35263	All sets are finite iff all ordinal sets are finite. (Contributed by BTernaryTau, 25-Jan-2026.)
⊢ (Fin = V ↔ ω = On)
Theorem	omprcomonb 35264	The class of all finite ordinals is a proper class iff all ordinal sets are finite. (Contributed by BTernaryTau, 25-Jan-2026.)
⊢ (¬ ω ∈ V ↔ ω = On)
Theorem	fineqvnttrclselem1 35265*	Lemma for fineqvnttrclse 35268. (Contributed by BTernaryTau, 12-Jan-2026.)
⊢ (𝐵 ∈ (ω ∖ 1o) → ∪ {𝑑 ∈ On ∣ (𝐴 +o 𝑑) = 𝐵} ∈ ω)
Theorem	fineqvnttrclselem2 35266*	Lemma for fineqvnttrclse 35268. (Contributed by BTernaryTau, 12-Jan-2026.)
⊢ 𝐹 = (𝑣 ∈ suc suc 𝑁 ↦ ∪ {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵})    ⇒   ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 +o (𝐹‘𝐴)) = 𝐵)
Theorem	fineqvnttrclselem3 35267*	Lemma for fineqvnttrclse 35268. (Contributed by BTernaryTau, 12-Jan-2026.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = suc 𝑦)}    &   ⊢ 𝐴 = ω    &   ⊢ 𝐹 = (𝑣 ∈ suc suc 𝑁 ↦ ∪ {𝑑 ∈ On ∣ (𝑣 +o 𝑑) = 𝐵})    ⇒   ⊢ ((𝐵 ∈ (ω ∖ 1o) ∧ 𝑁 ∈ 𝐵) → ∀𝑎 ∈ suc 𝑁(𝐹‘𝑎)𝑅(𝐹‘suc 𝑎))
Theorem	fineqvnttrclse 35268*	A counterexample demonstrating that ttrclse 9648 does not hold when all sets are finite. (Contributed by BTernaryTau, 12-Jan-2026.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = suc 𝑦)}    &   ⊢ 𝐴 = ω    ⇒   ⊢ (Fin = V → (𝑅 Se 𝐴 ∧ ¬ t++(𝑅 ↾ 𝐴) Se 𝐴))
Theorem	fineqvinfep 35269*	A counterexample demonstrating that tz9.1 9650 does not hold when all sets are finite and an infinite descending ∈-chain exists. (Contributed by BTernaryTau, 18-Feb-2026.)
⊢ 𝐴 = {(𝐹‘∅)}    ⇒   ⊢ ((Fin = V ∧ 𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ¬ ∃𝑦(𝐴 ⊆ 𝑦 ∧ Tr 𝑦))
21.5.2.2  Introduce ax-regs
Axiom	ax-regs 35270*	A strong version of the Axiom of Regularity. It states that if there exists a set with property 𝜑, then there must exist a set with property 𝜑 such that none of its elements have property 𝜑. This axiom can be derived from the axioms of ZF set theory as shown in axregs 35283, but this derivation relies on ax-inf2 9562 and is thus not possible in a finitist context. (Contributed by BTernaryTau, 29-Dec-2025.)
⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
Theorem	axreg 35271*	Derivation of ax-reg 9507 from ax-regs 35270 and Tarski's FOL axiom schemes. This demonstrates the sense in which ax-regs 35270 is a stronger version of ax-reg 9507. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
Theorem	axregscl 35272*	A version of ax-regs 35270 with a class variable instead of a wff variable. Axiom D in Gödel, The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (1940), p. 6. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)))
Theorem	axregszf 35273*	Derivation of zfregs 9653 using ax-regs 35270. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
Theorem	setindregs 35274*	Set (epsilon) induction. This version of setind 9668 replaces zfregs 9653 with axregszf 35273. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)
Theorem	setinds2regs 35275*	Principle of set induction (or E-induction). If a property passes from all elements of 𝑥 to 𝑥 itself, then it holds for all 𝑥. (Contributed by BTernaryTau, 31-Dec-2025.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	noinfepfnregs 35276*	There are no infinite descending ∈-chains, proven using ax-regs 35270. (Contributed by BTernaryTau, 18-Feb-2026.)
⊢ (𝐹 Fn ω → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
Theorem	noinfepregs 35277*	There are no infinite descending ∈-chains, proven using ax-regs 35270. (Contributed by BTernaryTau, 18-Feb-2026.)
⊢ ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)
Theorem	tz9.1regs 35278*	Every set has a transitive closure (the smallest transitive extension). This version of tz9.1 9650 depends on ax-regs 35270 instead of ax-reg 9507 and ax-inf2 9562. This suggests a possible answer to the third question posed in tz9.1 9650, namely that the missing property is that countably infinite classes must obey regularity. In ZF set theory we can prove this by showing that countably infinite classes are sets and thus ax-reg 9507 applies to them directly, but in a finitist context it seems that an axiom like ax-regs 35270 is required since countably infinite classes are proper classes. A related candidate for the missing property is the non-existence of infinite descending ∈-chains, proven as noinfep 9581 using ax-reg 9507 and ax-inf2 9562 and as noinfepregs 35277 using ax-regs 35270. If all sets are finite, then the existence of such a chain implies there is a set which does not have a transitive closure, as shown in fineqvinfep 35269. (Contributed by BTernaryTau, 31-Dec-2025.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))
Theorem	unir1regs 35279	The cumulative hierarchy of sets covers the universe. This version of unir1 9737 replaces setind 9668 with setindregs 35274. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ ∪ (𝑅1 “ On) = V
Theorem	trssfir1omregs 35280	If every element in a transitive class is finite, then every element is also hereditarily finite. This version of trssfir1om 35255 replaces setinds2 9672 with setinds2regs 35275. (Contributed by BTernaryTau, 20-Jan-2026.)
⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ ∪ (𝑅1 “ ω))
Theorem	r1omhfbregs 35281*	The class of all hereditarily finite sets is the only class with the property that all sets are members of it iff they are finite and all of their elements are members of it. This version of r1omhfb 35256 replaces setinds2 9672 with setinds2regs 35275 and trssfir1om 35255 with trssfir1omregs 35280. (Contributed by BTernaryTau, 21-Jan-2026.)
⊢ (𝐻 = ∪ (𝑅1 “ ω) ↔ ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
Theorem	fineqvr1ombregs 35282	All sets are finite iff all sets are hereditarily finite. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (Fin = V ↔ ∪ (𝑅1 “ ω) = V)
21.5.2.3  Derive ax-regs
Theorem	axregs 35283*	Derivation of ax-regs 35270 from the axioms of ZF set theory. (Contributed by BTernaryTau, 29-Dec-2025.)
⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))
21.5.2.4  Global choice
Theorem	gblacfnacd 35284*	If 𝐺 is a global choice function, then the Axiom of Choice (in the form of the right-hand side of dfac4 10044) holds. Note that 𝐺 must be a proper class by fndmexb 7857. This means we cannot show that the existence of a class that behaves as a global choice function is sufficient because we only have existential quantifiers for sets, not (proper) classes. However, if a class variant of exlimiv 1932 were available, then it could be used alongside the closed form of this theorem to prove that result. (Contributed by BTernaryTau, 12-Dec-2024.)
⊢ (𝜑 → 𝐺 Fn V)    &   ⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))    ⇒   ⊢ (𝜑 → ∀𝑥∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
Theorem	onvf1odlem1 35285*	Lemma for onvf1od 35289. (Contributed by BTernaryTau, 2-Dec-2025.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)
Theorem	onvf1odlem2 35286*	Lemma for onvf1od 35289. (Contributed by BTernaryTau, 2-Dec-2025.)
⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))    &   ⊢ 𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}    &   ⊢ 𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ 𝐴))    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝑉 → 𝑁 ∈ ((𝑅1‘𝑀) ∖ 𝐴)))
Theorem	onvf1odlem3 35287*	Lemma for onvf1od 35289. The value of 𝐹 at an ordinal 𝐴. (Contributed by BTernaryTau, 2-Dec-2025.)
⊢ 𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤}    &   ⊢ 𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ ran 𝑤))    &   ⊢ 𝐹 = recs((𝑤 ∈ V ↦ 𝑁))    &   ⊢ 𝐵 = ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝐴)}    &   ⊢ 𝐶 = (𝐺‘((𝑅1‘𝐵) ∖ (𝐹 “ 𝐴)))    ⇒   ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = 𝐶)
Theorem	onvf1odlem4 35288*	Lemma for onvf1od 35289. If the range of 𝐹 does not exist, then it must equal the universe. (Contributed by BTernaryTau, 4-Dec-2025.)
⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))    &   ⊢ 𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤}    &   ⊢ 𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ ran 𝑤))    &   ⊢ 𝐹 = recs((𝑤 ∈ V ↦ 𝑁))    &   ⊢ 𝐵 = ∩ {𝑢 ∈ On ∣ ∃𝑣 ∈ (𝑅1‘𝑢) ¬ 𝑣 ∈ (𝐹 “ 𝑡)}    &   ⊢ 𝐶 = (𝐺‘((𝑅1‘𝐵) ∖ (𝐹 “ 𝑡)))    ⇒   ⊢ (𝜑 → (¬ ran 𝐹 ∈ V → ran 𝐹 = V))
Theorem	onvf1od 35289*	If 𝐺 is a global choice function, then 𝐹 is a bijection from the ordinals to the universe. This is the ZFC version of (1 → 2) in https://tinyurl.com/hamkins-gblac. (Contributed by BTernaryTau, 5-Dec-2025.)
⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))    &   ⊢ 𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ ran 𝑤}    &   ⊢ 𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ ran 𝑤))    &   ⊢ 𝐹 = recs((𝑤 ∈ V ↦ 𝑁))    ⇒   ⊢ (𝜑 → 𝐹:On–1-1-onto→V)
Theorem	vonf1owev 35290*	If 𝐹 is a bijection from the universe to the ordinals, then 𝑅 well-orders the universe. This is the ZFC version of (2 → 3) in https://tinyurl.com/hamkins-gblac. (Contributed by BTernaryTau, 6-Dec-2025.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)}    ⇒   ⊢ (𝐹:V–1-1-onto→On → 𝑅 We V)
Theorem	wevgblacfn 35291*	If 𝑅 is a well-ordering of the universe, then 𝐺 is a global choice function. Here 𝐺 maps each set 𝑧 to its minimal element with respect to 𝑅 (except when 𝑧 is the empty set, in which case it is mapped to the empty set, though this is only done for convenience). This is the ZFC version of (3 → 1) in https://tinyurl.com/hamkins-gblac. (Contributed by BTernaryTau, 29-Jun-2025.)
⊢ 𝐺 = (𝑧 ∈ V ↦ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦})    ⇒   ⊢ (𝑅 We V → (𝐺 Fn V ∧ ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧)))
21.5.3  Real and complex numbers
Theorem	zltp1ne 35292	Integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 + 1) < 𝐵 ↔ (𝐴 < 𝐵 ∧ 𝐵 ≠ (𝐴 + 1))))
Theorem	nnltp1ne 35293	Positive integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 1) < 𝐵 ↔ (𝐴 < 𝐵 ∧ 𝐵 ≠ (𝐴 + 1))))
Theorem	nn0ltp1ne 35294	Nonnegative integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴 + 1) < 𝐵 ↔ (𝐴 < 𝐵 ∧ 𝐵 ≠ (𝐴 + 1))))
Theorem	0nn0m1nnn0 35295	A number is zero if and only if it's a nonnegative integer that becomes negative after subtracting 1. (Contributed by BTernaryTau, 30-Sep-2023.)
⊢ (𝑁 = 0 ↔ (𝑁 ∈ ℕ0 ∧ ¬ (𝑁 − 1) ∈ ℕ0))
Theorem	f1resfz0f1d 35296	If a function with a sequence of nonnegative integers (starting at 0) as its domain is one-to-one when 0 is removed, and if the range of that restriction does not contain the function's value at the removed integer, then the function is itself one-to-one. (Contributed by BTernaryTau, 4-Oct-2023.)
⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐹:(0...𝐾)⟶𝑉)    &   ⊢ (𝜑 → (𝐹 ↾ (1...𝐾)):(1...𝐾)–1-1→𝑉)    &   ⊢ (𝜑 → ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ∅)    ⇒   ⊢ (𝜑 → 𝐹:(0...𝐾)–1-1→𝑉)
Theorem	fisshasheq 35297	A finite set is equal to its subset if they are the same size. (Contributed by BTernaryTau, 3-Oct-2023.)
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ (♯‘𝐴) = (♯‘𝐵)) → 𝐴 = 𝐵)
Theorem	revpfxsfxrev 35298	The reverse of a prefix of a word is equal to the same-length suffix of the reverse of that word. (Contributed by BTernaryTau, 2-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (reverse‘(𝑊 prefix 𝐿)) = ((reverse‘𝑊) substr ⟨((♯‘𝑊) − 𝐿), (♯‘𝑊)⟩))
Theorem	swrdrevpfx 35299	A subword expressed in terms of reverses and prefixes. (Contributed by BTernaryTau, 3-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 substr ⟨𝐹, 𝐿⟩) = (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿 − 𝐹))))
21.5.4  Graph theory
Theorem	lfuhgr 35300*	A hypergraph is loop-free if and only if every edge connects at least two vertices. (Contributed by BTernaryTau, 15-Oct-2023.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))