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Theorem wdomac 9926
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* 𝑌𝑋𝑌)

Proof of Theorem wdomac
StepHypRef Expression
1 relwdom 9006 . . 3 Rel ≼*
21brrelex2i 5582 . 2 (𝑋* 𝑌𝑌 ∈ V)
3 reldom 8490 . . 3 Rel ≼
43brrelex2i 5582 . 2 (𝑋𝑌𝑌 ∈ V)
5 numth3 9869 . . 3 (𝑌 ∈ V → 𝑌 ∈ dom card)
6 wdomnumr 9467 . . 3 (𝑌 ∈ dom card → (𝑋* 𝑌𝑋𝑌))
75, 6syl 17 . 2 (𝑌 ∈ V → (𝑋* 𝑌𝑋𝑌))
82, 4, 7pm5.21nii 383 1 (𝑋* 𝑌𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2115  Vcvv 3471   class class class wbr 5039  dom cdm 5528  cdom 8482  * cwdom 9004  cardccrd 9340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-ac2 9862
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-er 8264  df-map 8383  df-en 8485  df-dom 8486  df-sdom 8487  df-wdom 9005  df-card 9344  df-acn 9347  df-ac 9519
This theorem is referenced by: (None)
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