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Theorem dfac10b 10178
Description: Axiom of Choice equivalent: every set is equinumerous to an ordinal (quantifier-free short cryptic version alluded to in df-ac 10154). (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
dfac10b (CHOICE ↔ ( ≈ “ On) = V)

Proof of Theorem dfac10b
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3482 . . . . 5 𝑥 ∈ V
21elima 6085 . . . 4 (𝑥 ∈ ( ≈ “ On) ↔ ∃𝑦 ∈ On 𝑦𝑥)
32bicomi 224 . . 3 (∃𝑦 ∈ On 𝑦𝑥𝑥 ∈ ( ≈ “ On))
43albii 1816 . 2 (∀𝑥𝑦 ∈ On 𝑦𝑥 ↔ ∀𝑥 𝑥 ∈ ( ≈ “ On))
5 dfac10c 10177 . 2 (CHOICE ↔ ∀𝑥𝑦 ∈ On 𝑦𝑥)
6 eqv 3488 . 2 (( ≈ “ On) = V ↔ ∀𝑥 𝑥 ∈ ( ≈ “ On))
74, 5, 63bitr4i 303 1 (CHOICE ↔ ( ≈ “ On) = V)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1535   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478   class class class wbr 5148  cima 5692  Oncon0 6386  cen 8981  CHOICEwac 10153
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
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-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-en 8985  df-card 9977  df-ac 10154
This theorem is referenced by:  axac10  43022
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