wdomac - Metamath Proof Explorer

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Theorem wdomac 10362

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 9401	. . 3 ⊢ Rel ≼^*
2	1	brrelex2i 5662	. 2 ⊢ (𝑋 ≼^* 𝑌 → 𝑌 ∈ V)
3		reldom 8788	. . 3 ⊢ Rel ≼
4	3	brrelex2i 5662	. 2 ⊢ (𝑋 ≼ 𝑌 → 𝑌 ∈ V)
5		numth3 10305	. . 3 ⊢ (𝑌 ∈ V → 𝑌 ∈ dom card)
6		wdomnumr 9899	. . 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 379	1 ⊢ (𝑋 ≼^* 𝑌 ↔ 𝑋 ≼ 𝑌)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∈ wcel 2105 Vcvv 3440 class class class wbr 5086 dom cdm 5607 ≼ cdom 8780 ≼^* cwdom 9399 cardccrd 9770

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5223 ax-sep 5237 ax-nul 5244 ax-pow 5302 ax-pr 5366 ax-un 7629 ax-ac2 10298

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3442 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4471 df-pw 4546 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4850 df-int 4892 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5170 df-tr 5204 df-id 5506 df-eprel 5512 df-po 5520 df-so 5521 df-fr 5562 df-se 5563 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-isom 6474 df-riota 7273 df-ov 7319 df-oprab 7320 df-mpo 7321 df-1st 7877 df-2nd 7878 df-frecs 8145 df-wrecs 8176 df-recs 8250 df-er 8547 df-map 8666 df-en 8783 df-dom 8784 df-sdom 8785 df-wdom 9400 df-card 9774 df-acn 9777 df-ac 9951

This theorem is referenced by: (None)

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