wdomac - Metamath Proof Explorer

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Theorem wdomac 10521

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 9560	. . 3 ⊢ Rel ≼^*
2	1	brrelex2i 5726	. 2 ⊢ (𝑋 ≼^* 𝑌 → 𝑌 ∈ V)
3		reldom 8944	. . 3 ⊢ Rel ≼
4	3	brrelex2i 5726	. 2 ⊢ (𝑋 ≼ 𝑌 → 𝑌 ∈ V)
5		numth3 10464	. . 3 ⊢ (𝑌 ∈ V → 𝑌 ∈ dom card)
6		wdomnumr 10058	. . 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 205 ∈ wcel 2098 Vcvv 3468 class class class wbr 5141 dom cdm 5669 ≼ cdom 8936 ≼^* cwdom 9558 cardccrd 9929

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-ac2 10457

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-wdom 9559 df-card 9933 df-acn 9936 df-ac 10110

This theorem is referenced by: (None)

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