P. 95 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9401-9500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	infex 9401	An infimum is a set. (Contributed by AV, 3-Sep-2020.)
⊢ 𝑅 Or 𝐴    ⇒   ⊢ inf(𝐵, 𝐴, 𝑅) ∈ V
Theorem	infmin 9402*	The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by AV, 3-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶)    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Theorem	infmo 9403*	Any class 𝐵 has at most one infimum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by AV, 6-Oct-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
Theorem	infeu 9404*	An infimum is unique. (Contributed by AV, 6-Oct-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
Theorem	fimin2g 9405*	A finite set has a minimum under a total order. (Contributed by AV, 6-Oct-2020.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)
Theorem	fiming 9406*	A finite set has a minimum under a total order. (Contributed by AV, 6-Oct-2020.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑥𝑅𝑦))
Theorem	fiinfg 9407*	Lemma showing existence and closure of infimum of a finite set. (Contributed by AV, 6-Oct-2020.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐴 𝑧𝑅𝑦)))
Theorem	fiinf2g 9408*	A finite set satisfies the conditions to have an infimum. (Contributed by AV, 6-Oct-2020.)
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
Theorem	fiinfcl 9409	A nonempty finite set contains its infimum. (Contributed by AV, 3-Sep-2020.)
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
Theorem	infltoreq 9410	The infimum of a finite set is less than or equal to all the elements of the set. (Contributed by AV, 4-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 = inf(𝐵, 𝐴, 𝑅))    ⇒   ⊢ (𝜑 → (𝑆𝑅𝐶 ∨ 𝐶 = 𝑆))
Theorem	infpr 9411	The infimum of a pair. (Contributed by AV, 4-Sep-2020.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → inf({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐵𝑅𝐶, 𝐵, 𝐶))
Theorem	infsupprpr 9412	The infimum of a proper pair is less than the supremum of this pair. (Contributed by AV, 13-Mar-2023.)
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → inf({𝐵, 𝐶}, 𝐴, 𝑅)𝑅sup({𝐵, 𝐶}, 𝐴, 𝑅))
Theorem	infsn 9413	The infimum of a singleton. (Contributed by NM, 2-Oct-2007.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵}, 𝐴, 𝑅) = 𝐵)
Theorem	inf00 9414	The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
⊢ inf(𝐵, ∅, 𝑅) = ∅
Theorem	infempty 9415*	The infimum of an empty set under a base set which has a unique greatest element is the greatest element of the base set. (Contributed by AV, 4-Sep-2020.)
⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) → inf(∅, 𝐴, 𝑅) = 𝑋)
Theorem	infiso 9416*	Image of an infimum under an isomorphism. (Contributed by AV, 4-Sep-2020.)
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦)))    &   ⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → inf((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))
2.4.39  Ordinal isomorphism, Hartogs's theorem
Syntax	coi 9417	Extend class definition to include the canonical order isomorphism to an ordinal.
class OrdIso(𝑅, 𝐴)
Definition	df-oi 9418*	Define the canonical order isomorphism from the well-order 𝑅 on 𝐴 to an ordinal. (Contributed by Mario Carneiro, 23-May-2015.)
⊢ OrdIso(𝑅, 𝐴) = if((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴), (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑥)𝑧𝑅𝑡}), ∅)
Theorem	dfoi 9419*	Rewrite df-oi 9418 with abbreviations. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}    &   ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))    &   ⊢ 𝐹 = recs(𝐺)    ⇒   ⊢ OrdIso(𝑅, 𝐴) = if((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴), (𝐹 ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}), ∅)
Theorem	oieq1 9420	Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
⊢ (𝑅 = 𝑆 → OrdIso(𝑅, 𝐴) = OrdIso(𝑆, 𝐴))
Theorem	oieq2 9421	Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
⊢ (𝐴 = 𝐵 → OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐵))
Theorem	nfoi 9422	Hypothesis builder for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥OrdIso(𝑅, 𝐴)
Theorem	ordiso2 9423	Generalize ordiso 9424 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵)
Theorem	ordiso 9424*	Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 ↔ ∃𝑓 𝑓 Isom E , E (𝐴, 𝐵)))
Theorem	ordtypecbv 9425*	Lemma for ordtype 9440. (Contributed by Mario Carneiro, 26-Jun-2015.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}    &   ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))    ⇒   ⊢ recs((𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) = 𝐹
Theorem	ordtypelem1 9426*	Lemma for ordtype 9440. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}    &   ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))    &   ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}    &   ⊢ 𝑂 = OrdIso(𝑅, 𝐴)    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    ⇒   ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇))
Theorem	ordtypelem2 9427*	Lemma for ordtype 9440. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}    &   ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))    &   ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}    &   ⊢ 𝑂 = OrdIso(𝑅, 𝐴)    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    ⇒   ⊢ (𝜑 → Ord 𝑇)
Theorem	ordtypelem3 9428*	Lemma for ordtype 9440. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}    &   ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))    &   ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}    &   ⊢ 𝑂 = OrdIso(𝑅, 𝐴)    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑀) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
Theorem	ordtypelem4 9429*	Lemma for ordtype 9440. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}    &   ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))    &   ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}    &   ⊢ 𝑂 = OrdIso(𝑅, 𝐴)    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    ⇒   ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
Theorem	ordtypelem5 9430*	Lemma for ordtype 9440. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}    &   ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))    &   ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}    &   ⊢ 𝑂 = OrdIso(𝑅, 𝐴)    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    ⇒   ⊢ (𝜑 → (Ord dom 𝑂 ∧ 𝑂:dom 𝑂⟶𝐴))
Theorem	ordtypelem6 9431*	Lemma for ordtype 9440. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}    &   ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))    &   ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}    &   ⊢ 𝑂 = OrdIso(𝑅, 𝐴)    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → (𝑁 ∈ 𝑀 → (𝑂‘𝑁)𝑅(𝑂‘𝑀)))
Theorem	ordtypelem7 9432*	Lemma for ordtype 9440. ran 𝑂 is an initial segment of 𝐴 under the well-order 𝑅. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}    &   ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))    &   ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}    &   ⊢ 𝑂 = OrdIso(𝑅, 𝐴)    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    ⇒   ⊢ (((𝜑 ∧ 𝑁 ∈ 𝐴) ∧ 𝑀 ∈ dom 𝑂) → ((𝑂‘𝑀)𝑅𝑁 ∨ 𝑁 ∈ ran 𝑂))
Theorem	ordtypelem8 9433*	Lemma for ordtype 9440. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}    &   ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))    &   ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}    &   ⊢ 𝑂 = OrdIso(𝑅, 𝐴)    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    ⇒   ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
Theorem	ordtypelem9 9434*	Lemma for ordtype 9440. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 9438 implies that either ran 𝑂 ⊆ 𝐴 is a proper class or dom 𝑂 = On. (Contributed by Mario Carneiro, 25-Jun-2015.) (Revised by AV, 28-Jul-2024.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}    &   ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))    &   ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}    &   ⊢ 𝑂 = OrdIso(𝑅, 𝐴)    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
Theorem	ordtypelem10 9435*	Lemma for ordtype 9440. Using ax-rep 5212, exclude the possibility that 𝑂 is a proper class and does not enumerate all of 𝐴. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ 𝐹 = recs(𝐺)    &   ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}    &   ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))    &   ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}    &   ⊢ 𝑂 = OrdIso(𝑅, 𝐴)    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    ⇒   ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
Theorem	oi0 9436	Definition of the ordinal isomorphism when its arguments are not meaningful. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ 𝐹 = OrdIso(𝑅, 𝐴)    ⇒   ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 = ∅)
Theorem	oicl 9437	The order type of the well-order 𝑅 on 𝐴 is an ordinal. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
⊢ 𝐹 = OrdIso(𝑅, 𝐴)    ⇒   ⊢ Ord dom 𝐹
Theorem	oif 9438	The order isomorphism of the well-order 𝑅 on 𝐴 is a function. (Contributed by Mario Carneiro, 23-May-2015.)
⊢ 𝐹 = OrdIso(𝑅, 𝐴)    ⇒   ⊢ 𝐹:dom 𝐹⟶𝐴
Theorem	oiiso2 9439	The order isomorphism of the well-order 𝑅 on 𝐴 is an isomorphism onto ran 𝑂 (which is a subset of 𝐴 by oif 9438). (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ 𝐹 = OrdIso(𝑅, 𝐴)    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, ran 𝐹))
Theorem	ordtype 9440	For any set-like well-ordered class, there is an isomorphic ordinal number called its order type. (Contributed by Jeff Hankins, 17-Oct-2009.) (Revised by Mario Carneiro, 25-Jun-2015.)
⊢ 𝐹 = OrdIso(𝑅, 𝐴)    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))
Theorem	oiiniseg 9441	ran 𝐹 is an initial segment of 𝐴 under the well-order 𝑅. (Contributed by Mario Carneiro, 26-Jun-2015.)
⊢ 𝐹 = OrdIso(𝑅, 𝐴)    ⇒   ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑁 ∈ 𝐴 ∧ 𝑀 ∈ dom 𝐹)) → ((𝐹‘𝑀)𝑅𝑁 ∨ 𝑁 ∈ ran 𝐹))
Theorem	ordtype2 9442	For any set-like well-ordered class, if the order isomorphism exists (is a set), then it maps some ordinal onto 𝐴 isomorphically. Otherwise, 𝐹 is a proper class, which implies that either ran 𝐹 ⊆ 𝐴 is a proper class or dom 𝐹 = On. This weak version of ordtype 9440 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ 𝐹 = OrdIso(𝑅, 𝐴)    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ 𝐹 ∈ V) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))
Theorem	oiexg 9443	The order isomorphism on a set is a set. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ 𝐹 = OrdIso(𝑅, 𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V)
Theorem	oion 9444	The order type of the well-order 𝑅 on 𝐴 is an ordinal. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 23-May-2015.)
⊢ 𝐹 = OrdIso(𝑅, 𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → dom 𝐹 ∈ On)
Theorem	oiiso 9445	The order isomorphism of the well-order 𝑅 on 𝐴 is an isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
⊢ 𝐹 = OrdIso(𝑅, 𝐴)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))
Theorem	oien 9446	The order type of a well-ordered set is equinumerous to the set. (Contributed by Mario Carneiro, 23-May-2015.)
⊢ 𝐹 = OrdIso(𝑅, 𝐴)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → dom 𝐹 ≈ 𝐴)
Theorem	oieu 9447	Uniqueness of the unique ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
⊢ 𝐹 = OrdIso(𝑅, 𝐴)    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ((Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 (𝐵, 𝐴)) ↔ (𝐵 = dom 𝐹 ∧ 𝐺 = 𝐹)))
Theorem	oismo 9448	When 𝐴 is a subclass of On, 𝐹 is a strictly monotone ordinal functions, and it is also complete (it is an isomorphism onto all of 𝐴). The proof avoids ax-rep 5212 (the second statement is trivial under ax-rep 5212). (Contributed by Mario Carneiro, 26-Jun-2015.)
⊢ 𝐹 = OrdIso( E , 𝐴)    ⇒   ⊢ (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴))
Theorem	oiid 9449	The order type of an ordinal under the ∈ order is itself, and the order isomorphism is the identity function. (Contributed by Mario Carneiro, 26-Jun-2015.)
⊢ (Ord 𝐴 → OrdIso( E , 𝐴) = ( I ↾ 𝐴))
Theorem	hartogslem1 9450*	Lemma for hartogs 9452. (Contributed by Mario Carneiro, 14-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
⊢ 𝐹 = {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}    &   ⊢ 𝑅 = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑓‘𝑤) ∧ 𝑡 = (𝑓‘𝑧)) ∧ 𝑤 E 𝑧)}    ⇒   ⊢ (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴}))
Theorem	hartogslem2 9451*	Lemma for hartogs 9452. (Contributed by Mario Carneiro, 14-Jan-2013.)
⊢ 𝐹 = {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}    &   ⊢ 𝑅 = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑓‘𝑤) ∧ 𝑡 = (𝑓‘𝑧)) ∧ 𝑤 E 𝑧)}    ⇒   ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V)
Theorem	hartogs 9452*	The class of ordinals dominated by a given set is an ordinal. A shorter (when taking into account lemmas hartogslem1 9450 and hartogslem2 9451) proof can be given using the axiom of choice, see ondomon 10476. As its label indicates, this result is used to justify the definition of the Hartogs function df-har 9465. (Contributed by Jeff Hankins, 22-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On)
Theorem	wofib 9453	The only sets which are well-ordered forwards and backwards are finite sets. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 23-May-2015.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) ↔ (𝑅 We 𝐴 ∧ ◡𝑅 We 𝐴))
Theorem	wemaplem1 9454*	Value of the lexicographic order on a sequence space. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑄 ∈ 𝑊) → (𝑃𝑇𝑄 ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑄‘𝑎) ∧ ∀𝑏 ∈ 𝐴 (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏)))))
Theorem	wemaplem2 9455*	Lemma for wemapso 9459. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Revised by AV, 21-Jul-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝑃 ∈ (𝐵 ↑m 𝐴))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m 𝐴))    &   ⊢ (𝜑 → 𝑄 ∈ (𝐵 ↑m 𝐴))    &   ⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝑆 Po 𝐵)    &   ⊢ (𝜑 → 𝑎 ∈ 𝐴)    &   ⊢ (𝜑 → (𝑃‘𝑎)𝑆(𝑋‘𝑎))    &   ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))    &   ⊢ (𝜑 → 𝑏 ∈ 𝐴)    &   ⊢ (𝜑 → (𝑋‘𝑏)𝑆(𝑄‘𝑏))    &   ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)))    ⇒   ⊢ (𝜑 → 𝑃𝑇𝑄)
Theorem	wemaplem3 9456*	Lemma for wemapso 9459. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Revised by AV, 21-Jul-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝑃 ∈ (𝐵 ↑m 𝐴))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m 𝐴))    &   ⊢ (𝜑 → 𝑄 ∈ (𝐵 ↑m 𝐴))    &   ⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝑆 Po 𝐵)    &   ⊢ (𝜑 → 𝑃𝑇𝑋)    &   ⊢ (𝜑 → 𝑋𝑇𝑄)    ⇒   ⊢ (𝜑 → 𝑃𝑇𝑄)
Theorem	wemappo 9457*	Construct lexicographic order on a function space based on a well-ordering of the indices and a total ordering of the values. Without totality on the values or least differing indices, the best we can prove here is a partial order. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by AV, 21-Jul-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ ((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) → 𝑇 Po (𝐵 ↑m 𝐴))
Theorem	wemapsolem 9458*	Lemma for wemapso 9459. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 21-Jul-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ 𝑈 ⊆ (𝐵 ↑m 𝐴)    &   ⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝑆 Or 𝐵)    &   ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈) ∧ 𝑎 ≠ 𝑏)) → ∃𝑐 ∈ dom (𝑎 ∖ 𝑏)∀𝑑 ∈ dom (𝑎 ∖ 𝑏) ¬ 𝑑𝑅𝑐)    ⇒   ⊢ (𝜑 → 𝑇 Or 𝑈)
Theorem	wemapso 9459*	Construct lexicographic order on a function space based on a well-ordering of the indices and a total ordering of the values. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 21-Jul-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐵 ↑m 𝐴))
Theorem	wemapso2lem 9460*	Lemma for wemapso2 9461. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 1-Jul-2019.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ 𝑈 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    ⇒   ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) ∧ 𝑍 ∈ 𝑊) → 𝑇 Or 𝑈)
Theorem	wemapso2 9461*	An alternative to having a well-order on 𝑅 in wemapso 9459 is to restrict the function set to finitely-supported functions. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 1-Jul-2019.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ 𝑈 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or 𝑈)
Theorem	card2on 9462*	The alternate definition of the cardinal of a set given in cardval2 9906 always gives a set, and indeed an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.)
⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On
Theorem	card2inf 9463*	The alternate definition of the cardinal of a set given in cardval2 9906 has the curious property that for non-numerable sets (for which ndmfv 6866 yields ∅), it still evaluates to a nonempty set, and indeed it contains ω. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴})
2.4.40  Hartogs function
Syntax	char 9464	Class symbol for the Hartogs function.
class har
Definition	df-har 9465*	Define the Hartogs function as mapping a set to the class of ordinals it dominates. That class is an ordinal by hartogs 9452, which is used in harf 9466. The Hartogs number of a set is the least ordinal not dominated by that set. Theorem harval2 9912 proves that the Hartogs function actually gives the Hartogs number for well-orderable sets. The Hartogs number of an ordinal is its cardinal successor. This is proved for finite ordinal in harsucnn 9913. Traditionally, the Hartogs number of a set 𝑋 is written ℵ(𝑋), and its cardinal successor, 𝑋 +; we use functional notation for this, and cannot use the aleph symbol because it is taken for the enumerating function of the infinite initial ordinals df-aleph 9855. Some authors define the Hartogs number of a set to be the least *infinite* ordinal which does not inject into it, thus causing the range to consist only of alephs. We use the simpler definition where the value can be any successor cardinal. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥})
Theorem	harf 9466	Functionality of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ har:V⟶On
Theorem	harcl 9467	Values of the Hartogs function are ordinals (closure of the Hartogs function in the ordinals). (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ (har‘𝑋) ∈ On
Theorem	harval 9468*	Function value of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ (𝑋 ∈ 𝑉 → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋})
Theorem	elharval 9469	The Hartogs number of a set contains exactly the ordinals that set dominates. Combined with harcl 9467, this implies that the Hartogs number of a set is greater than all ordinals that set dominates. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.)
⊢ (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋))
Theorem	harndom 9470	The Hartogs number of a set does not inject into that set. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.)
⊢ ¬ (har‘𝑋) ≼ 𝑋
Theorem	harword 9471	Weak ordering property of the Hartogs function. (Contributed by Stefan O'Rear, 14-Feb-2015.)
⊢ (𝑋 ≼ 𝑌 → (har‘𝑋) ⊆ (har‘𝑌))
2.4.41  Weak dominance
Syntax	cwdom 9472	Class symbol for the weak dominance relation.
class ≼*
Definition	df-wdom 9473*	A set is weakly dominated by a "larger" set if the "larger" set can be mapped onto the "smaller" set or the smaller set is empty, or equivalently, if the smaller set can be placed into bijection with some partition of the larger set. Dominance (df-dom 8888) implies weak dominance (over ZF). The principle asserting the converse is known as the partition principle and is independent of ZF. Theorem fodom 10436 proves that the axiom of choice implies the partition principle (over ZF). It is not known whether the partition principle is equivalent to the axiom of choice (over ZF), although it is know to imply dependent choice. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ ≼* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}
Theorem	relwdom 9474	Weak dominance is a relation. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ Rel ≼*
Theorem	brwdom 9475*	Property of weak dominance (definitional form). (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ (𝑌 ∈ 𝑉 → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
Theorem	brwdomi 9476*	Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015.)
⊢ (𝑋 ≼* 𝑌 → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))
Theorem	brwdomn0 9477*	Weak dominance over nonempty sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
Theorem	0wdom 9478	Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ (𝑋 ∈ 𝑉 → ∅ ≼* 𝑋)
Theorem	fowdom 9479	An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ≼* 𝑌)
Theorem	wdomref 9480	Reflexivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ (𝑋 ∈ 𝑉 → 𝑋 ≼* 𝑋)
Theorem	brwdom2 9481*	Alternate characterization of the weak dominance predicate which does not require special treatment of the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ (𝑌 ∈ 𝑉 → (𝑋 ≼* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))
Theorem	domwdom 9482	Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.)
⊢ (𝑋 ≼ 𝑌 → 𝑋 ≼* 𝑌)
Theorem	wdomtr 9483	Transitivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
⊢ ((𝑋 ≼* 𝑌 ∧ 𝑌 ≼* 𝑍) → 𝑋 ≼* 𝑍)
Theorem	wdomen1 9484	Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼* 𝐶 ↔ 𝐵 ≼* 𝐶))
Theorem	wdomen2 9485	Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
⊢ (𝐴 ≈ 𝐵 → (𝐶 ≼* 𝐴 ↔ 𝐶 ≼* 𝐵))
Theorem	wdompwdom 9486	Weak dominance strengthens to usual dominance on the power sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
⊢ (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)
Theorem	canthwdom 9487	Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 9061, equivalent to canth 7314). (Contributed by Mario Carneiro, 15-May-2015.)
⊢ ¬ 𝒫 𝐴 ≼* 𝐴
Theorem	wdom2d 9488*	Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 5212). (Contributed by Stefan O'Rear, 13-Feb-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)    ⇒   ⊢ (𝜑 → 𝐴 ≼* 𝐵)
Theorem	wdomd 9489*	Deduce weak dominance from an implicit onto function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋)    ⇒   ⊢ (𝜑 → 𝐴 ≼* 𝐵)
Theorem	brwdom3 9490*	Condition for weak dominance with a condition reminiscent of wdomd 9489. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑋 ≼* 𝑌 ↔ ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)))
Theorem	brwdom3i 9491*	Weak dominance implies existence of a covering function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
⊢ (𝑋 ≼* 𝑌 → ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦))
Theorem	unwdomg 9492	Weak dominance of a (disjoint) union. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼* (𝐵 ∪ 𝐷))
Theorem	xpwdomg 9493	Weak dominance of a Cartesian product. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))
Theorem	wdomima2g 9494	A set is weakly dominant over its image under any function. This version of wdomimag 9495 is stated so as to avoid ax-rep 5212. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 “ 𝐴) ≼* 𝐴)
Theorem	wdomimag 9495	A set is weakly dominant over its image under any function. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ≼* 𝐴)
Theorem	unxpwdom2 9496	Lemma for unxpwdom 9497. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ ((𝐴 × 𝐴) ≈ (𝐵 ∪ 𝐶) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))
Theorem	unxpwdom 9497	If a Cartesian product is dominated by a union, then the base set is either weakly dominated by one factor of the union or dominated by the other. Extracted from Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))
Theorem	ixpiunwdom 9498*	Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 8869 this shows that ∪ 𝑥 ∈ 𝐴𝐵 and X𝑥 ∈ 𝐴𝐵 have closely linked cardinalities. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼* (X𝑥 ∈ 𝐴 𝐵 × 𝐴))
Theorem	harwdom 9499	The value of the Hartogs function at a set 𝑋 is weakly dominated by 𝒫 (𝑋 × 𝑋). This follows from a more precise analysis of the bound used in hartogs 9452 to prove that (har‘𝑋) is an ordinal. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (𝑋 ∈ 𝑉 → (har‘𝑋) ≼* 𝒫 (𝑋 × 𝑋))
2.5  ZF Set Theory - add the Axiom of Regularity
2.5.1  Introduce the Axiom of Regularity
Axiom	ax-reg 9500*	Axiom of Regularity. An axiom of Zermelo-Fraenkel set theory. Also called the Axiom of Foundation. A rather non-intuitive axiom that denies more than it asserts, it states (in the form of zfreg 9504) that every nonempty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself (elirrv 9505). A stronger version that works for proper classes is proved as zfregs 9644. (Contributed by NM, 14-Aug-1993.)
⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))