P. 63 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6201-6300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tfrcllemssrecs 6201*	Lemma for tfrcl 6213. The union of functions acceptable for tfrcl 6213 is a subset of recs. (Contributed by Jim Kingdon, 25-Mar-2022.)
⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}    &   ⊢ (𝜑 → Ord 𝑋)    ⇒   ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺))
Theorem	tfrcllemsucfn 6202*	We can extend an acceptable function by one element to produce a function. Lemma for tfrcl 6213. (Contributed by Jim Kingdon, 24-Mar-2022.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → Ord 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)    &   ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}    &   ⊢ (𝜑 → 𝑧 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑔:𝑧⟶𝑆)    &   ⊢ (𝜑 → 𝑔 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶𝑆)
Theorem	tfrcllemsucaccv 6203*	Lemma for tfrcl 6213. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 24-Mar-2022.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → Ord 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)    &   ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑧 ∈ 𝑌)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑔:𝑧⟶𝑆)    &   ⊢ (𝜑 → 𝑔 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐴)
Theorem	tfrcllembacc 6204*	Lemma for tfrcl 6213. Each element of 𝐵 is an acceptable function. (Contributed by Jim Kingdon, 25-Mar-2022.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → Ord 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)    &   ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}    &   ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
Theorem	tfrcllembxssdm 6205*	Lemma for tfrcl 6213. The union of 𝐵 is defined on all elements of 𝑋. (Contributed by Jim Kingdon, 25-Mar-2022.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → Ord 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)    &   ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}    &   ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))    ⇒   ⊢ (𝜑 → 𝐷 ⊆ dom ∪ 𝐵)
Theorem	tfrcllembfn 6206*	Lemma for tfrcl 6213. The union of 𝐵 is a function defined on 𝑥. (Contributed by Jim Kingdon, 25-Mar-2022.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → Ord 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)    &   ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}    &   ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))    ⇒   ⊢ (𝜑 → ∪ 𝐵:𝐷⟶𝑆)
Theorem	tfrcllembex 6207*	Lemma for tfrcl 6213. The set 𝐵 exists. (Contributed by Jim Kingdon, 25-Mar-2022.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → Ord 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)    &   ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}    &   ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))    ⇒   ⊢ (𝜑 → 𝐵 ∈ V)
Theorem	tfrcllemubacc 6208*	Lemma for tfrcl 6213. The union of 𝐵 satisfies the recursion rule. (Contributed by Jim Kingdon, 25-Mar-2022.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → Ord 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)    &   ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}    &   ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))    ⇒   ⊢ (𝜑 → ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))
Theorem	tfrcllemex 6209*	Lemma for tfrcl 6213. (Contributed by Jim Kingdon, 26-Mar-2022.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → Ord 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)    &   ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}    &   ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝐷⟶𝑆 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢))))
Theorem	tfrcllemaccex 6210*	We can define an acceptable function on any element of 𝑋. As with many of the transfinite recursion theorems, we have hypotheses that state that 𝐹 is a function and that it is defined up to 𝑋. (Contributed by Jim Kingdon, 26-Mar-2022.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → Ord 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)    &   ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑋) → ∃𝑔(𝑔:𝐶⟶𝑆 ∧ ∀𝑢 ∈ 𝐶 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
Theorem	tfrcllemres 6211*	Lemma for tfr1on 6199. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-Mar-2022.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → Ord 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)    &   ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝑌 ⊆ dom 𝐹)
Theorem	tfrcldm 6212*	Recursion is defined on an ordinal if the characteristic function satisfies a closure hypothesis up to a suitable point. (Contributed by Jim Kingdon, 26-Mar-2022.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → Ord 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ ∪ 𝑋)    ⇒   ⊢ (𝜑 → 𝑌 ∈ dom 𝐹)
Theorem	tfrcl 6213*	Closure for transfinite recursion. As with tfr1on 6199, the characteristic function must be defined up to a suitable point, not necessarily on all ordinals. (Contributed by Jim Kingdon, 25-Mar-2022.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → Ord 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ ∪ 𝑋)    ⇒   ⊢ (𝜑 → (𝐹‘𝑌) ∈ 𝑆)
Theorem	tfri1 6214*	Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition. The condition is that 𝐺 is defined "everywhere", which is stated here as (𝐺‘𝑥) ∈ V. Alternately, ∀𝑥 ∈ On∀𝑓(𝑓 Fn 𝑥 → 𝑓 ∈ dom 𝐺) would suffice. Given a function 𝐺 satisfying that condition, we define a class 𝐴 of all "acceptable" functions. The final function we're interested in is the union 𝐹 = recs(𝐺) of them. 𝐹 is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of 𝐹. In this first part we show that 𝐹 is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)    ⇒   ⊢ 𝐹 Fn On
Theorem	tfri2 6215*	Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule 𝐺 ( as described at tfri1 6214). Here we show that the function 𝐹 has the property that for any function 𝐺 satisfying that condition, the "next" value of 𝐹 is 𝐺 recursively applied to all "previous" values of 𝐹. (Contributed by Jim Kingdon, 4-May-2019.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)    ⇒   ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴)))
Theorem	tfri3 6216*	Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule 𝐺 ( as described at tfri1 6214). Finally, we show that 𝐹 is unique. We do this by showing that any class 𝐵 with the same properties of 𝐹 that we showed in parts 1 and 2 is identical to 𝐹. (Contributed by Jim Kingdon, 4-May-2019.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)    ⇒   ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐵 = 𝐹)
Theorem	tfrex 6217*	The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐹‘𝐴) ∈ V)
2.6.20  Recursive definition generator
Syntax	crdg 6218	Extend class notation with the recursive definition generator, with characteristic function 𝐹 and initial value 𝐼.
class rec(𝐹, 𝐼)
Definition	df-irdg 6219*	Define a recursive definition generator on On (the class of ordinal numbers) with characteristic function 𝐹 and initial value 𝐼. This rather amazing operation allows us to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. On the other hand, the elimination of a direct definition is a matter of simple mechanical substitution. The price paid is the daunting complexity of our rec operation (especially when df-recs 6154 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple. In classical logic it would be easier to divide this definition into cases based on whether the domain of 𝑔 is zero, a successor, or a limit ordinal. Cases do not (in general) work that way in intuitionistic logic, so instead we choose a definition which takes the union of all the results of the characteristic function for ordinals in the domain of 𝑔. This means that this definition has the expected properties for increasing and continuous ordinal functions, which include ordinal addition and multiplication. For finite recursion we also define df-frec 6240 and for suitable characteristic functions df-frec 6240 yields the same result as rec restricted to ω, as seen at frecrdg 6257. Note: We introduce rec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Jim Kingdon, 19-May-2019.)
⊢ rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ (𝐼 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
Theorem	rdgeq1 6220	Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐴) = rec(𝐺, 𝐴))
Theorem	rdgeq2 6221	Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
⊢ (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))
Theorem	rdgfun 6222	The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ Fun rec(𝐹, 𝐴)
Theorem	rdgtfr 6223*	The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.)
⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → (Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ V))
Theorem	rdgruledefgg 6224*	The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉) → (Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ V))
Theorem	rdgruledefg 6225*	The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
⊢ 𝐹 Fn V    ⇒   ⊢ (𝐴 ∈ 𝑉 → (Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ V))
Theorem	rdgexggg 6226	The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
Theorem	rdgexgg 6227	The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
⊢ 𝐹 Fn V    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
Theorem	rdgifnon 6228	The recursive definition generator is a function on ordinal numbers. The 𝐹 Fn V condition states that the characteristic function is defined for all sets (being defined for all ordinals might be enough if being used in a manner similar to rdgon 6235; in cases like df-oadd 6269 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.)
⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉) → rec(𝐹, 𝐴) Fn On)
Theorem	rdgifnon2 6229*	The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.)
⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → rec(𝐹, 𝐴) Fn On)
Theorem	rdgivallem 6230*	Value of the recursive definition generator. Lemma for rdgival 6231 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.)
⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))))
Theorem	rdgival 6231*	Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.)
⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))))
Theorem	rdgss 6232	Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.)
⊢ (𝜑 → 𝐹 Fn V)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐴) ⊆ (rec(𝐹, 𝐼)‘𝐵))
Theorem	rdgisuc1 6233*	One way of describing the value of the recursive definition generator at a successor. There is no condition on the characteristic function 𝐹 other than 𝐹 Fn V. Given that, the resulting expression encompasses both the expected successor term (𝐹‘(rec(𝐹, 𝐴)‘𝐵)) but also terms that correspond to the initial value 𝐴 and to limit ordinals ∪ 𝑥 ∈ 𝐵(𝐹‘(rec(𝐹, 𝐴)‘𝑥)). If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 6234. (Contributed by Jim Kingdon, 9-Jun-2019.)
⊢ (𝜑 → 𝐹 Fn V)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐴 ∪ (∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) ∪ (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))))
Theorem	rdgisucinc 6234*	Value of the recursive definition generator at a successor. This can be thought of as a generalization of oasuc 6312 and omsuc 6320. (Contributed by Jim Kingdon, 29-Aug-2019.)
⊢ (𝜑 → 𝐹 Fn V)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵)))
Theorem	rdgon 6235*	Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.) (Revised by Jim Kingdon, 13-Apr-2022.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ On)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)
Theorem	rdg0 6236	The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (rec(𝐹, 𝐴)‘∅) = 𝐴
Theorem	rdg0g 6237	The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
⊢ (𝐴 ∈ 𝐶 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
Theorem	rdgexg 6238	The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐹 Fn V    ⇒   ⊢ (𝐵 ∈ 𝑉 → (rec(𝐹, 𝐴)‘𝐵) ∈ V)
2.6.21  Finite recursion
Syntax	cfrec 6239	Extend class notation with the finite recursive definition generator, with characteristic function 𝐹 and initial value 𝐼.
class frec(𝐹, 𝐼)
Definition	df-frec 6240*	Define a recursive definition generator on ω (the class of finite ordinals) with characteristic function 𝐹 and initial value 𝐼. This rather amazing operation allows us to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. On the other hand, the elimination of a direct definition is a matter of simple mechanical substitution. The price paid is the daunting complexity of our frec operation (especially when df-recs 6154 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple; see frec0g 6246 and frecsuc 6256. Unlike with transfinite recursion, finite recurson can readily divide definitions and proofs into zero and successor cases, because even without excluded middle we have theorems such as nn0suc 4476. The analogous situation with transfinite recursion - being able to say that an ordinal is zero, successor, or limit - is enabled by excluded middle and thus is not available to us. For the characteristic functions which satisfy the conditions given at frecrdg 6257, this definition and df-irdg 6219 restricted to ω produce the same result. Note: We introduce frec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Mario Carneiro and Jim Kingdon, 10-Aug-2019.)
⊢ frec(𝐹, 𝐼) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))})) ↾ ω)
Theorem	freceq1 6241	Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝐹 = 𝐺 → frec(𝐹, 𝐴) = frec(𝐺, 𝐴))
Theorem	freceq2 6242	Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ (𝐴 = 𝐵 → frec(𝐹, 𝐴) = frec(𝐹, 𝐵))
Theorem	frecex 6243	Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.)
⊢ frec(𝐹, 𝐴) ∈ V
Theorem	frecfun 6244	Finite recursion produces a function. See also frecfnom 6250 which also states that the domain of that function is ω but which puts conditions on 𝐴 and 𝐹. (Contributed by Jim Kingdon, 13-Feb-2022.)
⊢ Fun frec(𝐹, 𝐴)
Theorem	nffrec 6245	Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥frec(𝐹, 𝐴)
Theorem	frec0g 6246	The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)
⊢ (𝐴 ∈ 𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
Theorem	frecabex 6247*	The class abstraction from df-frec 6240 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.)
⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑦(𝐹‘𝑦) ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    ⇒   ⊢ (𝜑 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V)
Theorem	frecabcl 6248*	The class abstraction from df-frec 6240 exists. Unlike frecabex 6247 the function 𝐹 only needs to be defined on 𝑆, not all sets. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 21-Mar-2022.)
⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → 𝐺:𝑁⟶𝑆)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    ⇒   ⊢ (𝜑 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝐺 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝐺‘𝑚))) ∨ (dom 𝐺 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ 𝑆)
Theorem	frectfr 6249*	Lemma to connect transfinite recursion theorems with finite recursion. That is, given the conditions 𝐹 Fn V and 𝐴 ∈ 𝑉 on frec(𝐹, 𝐴), we want to be able to apply tfri1d 6184 or tfri2d 6185, and this lemma lets us satisfy hypotheses of those theorems. (Contributed by Jim Kingdon, 15-Aug-2019.)
⊢ 𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})    ⇒   ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → ∀𝑦(Fun 𝐺 ∧ (𝐺‘𝑦) ∈ V))
Theorem	frecfnom 6250*	The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.)
⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → frec(𝐹, 𝐴) Fn ω)
Theorem	freccllem 6251*	Lemma for freccl 6252. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 27-Mar-2022.)
⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ ω)    &   ⊢ 𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))    ⇒   ⊢ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)
Theorem	freccl 6252*	Closure for finite recursion. (Contributed by Jim Kingdon, 27-Mar-2022.)
⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ ω)    ⇒   ⊢ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)
Theorem	frecfcllem 6253*	Lemma for frecfcl 6254. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 30-Mar-2022.)
⊢ 𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))    ⇒   ⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → frec(𝐹, 𝐴):ω⟶𝑆)
Theorem	frecfcl 6254*	Finite recursion yields a function on the natural numbers. (Contributed by Jim Kingdon, 30-Mar-2022.)
⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → frec(𝐹, 𝐴):ω⟶𝑆)
Theorem	frecsuclem 6255*	Lemma for frecsuc 6256. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 29-Mar-2022.)
⊢ 𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})    ⇒   ⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(frec(𝐹, 𝐴)‘𝐵)))
Theorem	frecsuc 6256*	The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 31-Mar-2022.)
⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(frec(𝐹, 𝐴)‘𝐵)))
Theorem	frecrdg 6257*	Transfinite recursion restricted to omega. Given a suitable characteristic function, df-frec 6240 produces the same results as df-irdg 6219 restricted to ω. Presumably the theorem would also hold if 𝐹 Fn V were changed to ∀𝑧(𝐹‘𝑧) ∈ V. (Contributed by Jim Kingdon, 29-Aug-2019.)
⊢ (𝜑 → 𝐹 Fn V)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑥 𝑥 ⊆ (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → frec(𝐹, 𝐴) = (rec(𝐹, 𝐴) ↾ ω))
2.6.22  Ordinal arithmetic
Syntax	c1o 6258	Extend the definition of a class to include the ordinal number 1.
class 1o
Syntax	c2o 6259	Extend the definition of a class to include the ordinal number 2.
class 2o
Syntax	c3o 6260	Extend the definition of a class to include the ordinal number 3.
class 3o
Syntax	c4o 6261	Extend the definition of a class to include the ordinal number 4.
class 4o
Syntax	coa 6262	Extend the definition of a class to include the ordinal addition operation.
class +o
Syntax	comu 6263	Extend the definition of a class to include the ordinal multiplication operation.
class ·o
Syntax	coei 6264	Extend the definition of a class to include the ordinal exponentiation operation.
class ↑o
Definition	df-1o 6265	Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)
⊢ 1o = suc ∅
Definition	df-2o 6266	Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)
⊢ 2o = suc 1o
Definition	df-3o 6267	Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ 3o = suc 2o
Definition	df-4o 6268	Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ 4o = suc 3o
Definition	df-oadd 6269*	Define the ordinal addition operation. (Contributed by NM, 3-May-1995.)
⊢ +o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦))
Definition	df-omul 6270*	Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)
⊢ ·o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +o 𝑥)), ∅)‘𝑦))
Definition	df-oexpi 6271*	Define the ordinal exponentiation operation. This definition is similar to a conventional definition of exponentiation except that it defines ∅ ↑o 𝐴 to be 1o for all 𝐴 ∈ On, in order to avoid having different cases for whether the base is ∅ or not. (Contributed by Mario Carneiro, 4-Jul-2019.)
⊢ ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
Theorem	1on 6272	Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)
⊢ 1o ∈ On
Theorem	1oex 6273	Ordinal 1 is a set. (Contributed by BJ, 4-Jul-2022.)
⊢ 1o ∈ V
Theorem	2on 6274	Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
⊢ 2o ∈ On
Theorem	2on0 6275	Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
⊢ 2o ≠ ∅
Theorem	3on 6276	Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ 3o ∈ On
Theorem	4on 6277	Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ 4o ∈ On
Theorem	df1o2 6278	Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
⊢ 1o = {∅}
Theorem	df2o3 6279	Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 2o = {∅, 1o}
Theorem	df2o2 6280	Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
⊢ 2o = {∅, {∅}}
Theorem	1n0 6281	Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
⊢ 1o ≠ ∅
Theorem	xp01disj 6282	Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1o})) = ∅
Theorem	xp01disjl 6283	Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)
⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐶)) = ∅
Theorem	ordgt0ge1 6284	Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴))
Theorem	ordge1n0im 6285	An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.)
⊢ (Ord 𝐴 → (1o ⊆ 𝐴 → 𝐴 ≠ ∅))
Theorem	el1o 6286	Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)
Theorem	dif1o 6287	Two ways to say that 𝐴 is a nonzero number of the set 𝐵. (Contributed by Mario Carneiro, 21-May-2015.)
⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))
Theorem	2oconcl 6288	Closure of the pair swapping function on 2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o)
Theorem	0lt1o 6289	Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
⊢ ∅ ∈ 1o
Theorem	0lt2o 6290	Ordinal zero is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
⊢ ∅ ∈ 2o
Theorem	1lt2o 6291	Ordinal one is less than ordinal two. (Contributed by Jim Kingdon, 31-Jul-2022.)
⊢ 1o ∈ 2o
Theorem	oafnex 6292	The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.)
⊢ (𝑥 ∈ V ↦ suc 𝑥) Fn V
Theorem	sucinc 6293*	Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.)
⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧)    ⇒   ⊢ ∀𝑥 𝑥 ⊆ (𝐹‘𝑥)
Theorem	sucinc2 6294*	Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.)
⊢ 𝐹 = (𝑧 ∈ V ↦ suc 𝑧)    ⇒   ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵))
Theorem	fnoa 6295	Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Mario Carneiro, 3-Jul-2019.)
⊢ +o Fn (On × On)
Theorem	oaexg 6296	Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 +o 𝐵) ∈ V)
Theorem	omfnex 6297*	The characteristic function for ordinal multiplication is defined everywhere. (Contributed by Jim Kingdon, 23-Aug-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) Fn V)
Theorem	fnom 6298	Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.)
⊢ ·o Fn (On × On)
Theorem	omexg 6299	Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ·o 𝐵) ∈ V)
Theorem	fnoei 6300	Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)
⊢ ↑o Fn (On × On)