wdomac - Metamath Proof Explorer

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Theorem wdomac 10561

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 9602	. . 3 ⊢ Rel ≼^*
2	1	brrelex2i 5731	. 2 ⊢ (𝑋 ≼^* 𝑌 → 𝑌 ∈ V)
3		reldom 8972	. . 3 ⊢ Rel ≼
4	3	brrelex2i 5731	. 2 ⊢ (𝑋 ≼ 𝑌 → 𝑌 ∈ V)
5		numth3 10504	. . 3 ⊢ (𝑌 ∈ V → 𝑌 ∈ dom card)
6		wdomnumr 10100	. . 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 377	1 ⊢ (𝑋 ≼^* 𝑌 ↔ 𝑋 ≼ 𝑌)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∈ wcel 2099 Vcvv 3462 class class class wbr 5145 dom cdm 5674 ≼ cdom 8964 ≼^* cwdom 9600 cardccrd 9971

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-ac2 10497

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-wdom 9601 df-card 9975 df-acn 9978 df-ac 10152

This theorem is referenced by: (None)

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