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Theorem dfac10b 9554
Description: Axiom of Choice equivalent: every set is equinumerous to an ordinal (quantifier-free short cryptic version alluded to in df-ac 9531). (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 3472 . . . . 5 𝑥 ∈ V
21elima 5912 . . . 4 (𝑥 ∈ ( ≈ “ On) ↔ ∃𝑦 ∈ On 𝑦𝑥)
32bicomi 227 . . 3 (∃𝑦 ∈ On 𝑦𝑥𝑥 ∈ ( ≈ “ On))
43albii 1821 . 2 (∀𝑥𝑦 ∈ On 𝑦𝑥 ↔ ∀𝑥 𝑥 ∈ ( ≈ “ On))
5 dfac10c 9553 . 2 (CHOICE ↔ ∀𝑥𝑦 ∈ On 𝑦𝑥)
6 eqv 3477 . 2 (( ≈ “ On) = V ↔ ∀𝑥 𝑥 ∈ ( ≈ “ On))
74, 5, 63bitr4i 306 1 (CHOICE ↔ ( ≈ “ On) = V)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536   = wceq 1538  wcel 2114  wrex 3131  Vcvv 3469   class class class wbr 5042  cima 5535  Oncon0 6169  cen 8493  CHOICEwac 9530
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-wrecs 7934  df-recs 7995  df-en 8497  df-card 9356  df-ac 9531
This theorem is referenced by:  axac10  39908
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