wdomac - Metamath Proof Explorer

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Theorem wdomac 10567

Description: When assuming AC, weak and usual dominance coincide. It is not known if this is an AC equivalent. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)

**Assertion**
Ref	Expression
wdomac	⊢ (𝑋 ≼^* 𝑌 ↔ 𝑋 ≼ 𝑌)

**Proof of Theorem wdomac**
Step	Hyp	Ref	Expression
1		relwdom 9606	. . 3 ⊢ Rel ≼^*
2	1	brrelex2i 5742	. 2 ⊢ (𝑋 ≼^* 𝑌 → 𝑌 ∈ V)
3		reldom 8991	. . 3 ⊢ Rel ≼
4	3	brrelex2i 5742	. 2 ⊢ (𝑋 ≼ 𝑌 → 𝑌 ∈ V)
5		numth3 10510	. . 3 ⊢ (𝑌 ∈ V → 𝑌 ∈ dom card)
6		wdomnumr 10104	. . 3 ⊢ (𝑌 ∈ dom card → (𝑋 ≼^* 𝑌 ↔ 𝑋 ≼ 𝑌))
7	5, 6	syl 17	. 2 ⊢ (𝑌 ∈ V → (𝑋 ≼^* 𝑌 ↔ 𝑋 ≼ 𝑌))
8	2, 4, 7	pm5.21nii 378	1 ⊢ (𝑋 ≼^* 𝑌 ↔ 𝑋 ≼ 𝑌)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 dom cdm 5685 ≼ cdom 8983 ≼^* cwdom 9604 cardccrd 9975

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-ac2 10503

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-wdom 9605 df-card 9979 df-acn 9982 df-ac 10156

This theorem is referenced by: (None)

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