wdomac - Metamath Proof Explorer

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Theorem wdomac 10565

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 9604	. . 3 ⊢ Rel ≼^*
2	1	brrelex2i 5746	. 2 ⊢ (𝑋 ≼^* 𝑌 → 𝑌 ∈ V)
3		reldom 8990	. . 3 ⊢ Rel ≼
4	3	brrelex2i 5746	. 2 ⊢ (𝑋 ≼ 𝑌 → 𝑌 ∈ V)
5		numth3 10508	. . 3 ⊢ (𝑌 ∈ V → 𝑌 ∈ dom card)
6		wdomnumr 10102	. . 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 2106 Vcvv 3478 class class class wbr 5148 dom cdm 5689 ≼ cdom 8982 ≼^* cwdom 9602 cardccrd 9973

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-ac2 10501

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-wdom 9603 df-card 9977 df-acn 9980 df-ac 10154

This theorem is referenced by: (None)

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