Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac10b Structured version   Visualization version   GIF version

Theorem dfac10b 8844
 Description: Axiom of Choice equivalent: every set is equinumerous to an ordinal (quantifier-free short cryptic version alluded to in df-ac 8822). (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 3176 . . . . 5 𝑥 ∈ V
21elima 5390 . . . 4 (𝑥 ∈ ( ≈ “ On) ↔ ∃𝑦 ∈ On 𝑦𝑥)
32bicomi 213 . . 3 (∃𝑦 ∈ On 𝑦𝑥𝑥 ∈ ( ≈ “ On))
43albii 1737 . 2 (∀𝑥𝑦 ∈ On 𝑦𝑥 ↔ ∀𝑥 𝑥 ∈ ( ≈ “ On))
5 dfac10c 8843 . 2 (CHOICE ↔ ∀𝑥𝑦 ∈ On 𝑦𝑥)
6 eqv 3178 . 2 (( ≈ “ On) = V ↔ ∀𝑥 𝑥 ∈ ( ≈ “ On))
74, 5, 63bitr4i 291 1 (CHOICE ↔ ( ≈ “ On) = V)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   class class class wbr 4583   “ cima 5041  Oncon0 5640   ≈ cen 7838  CHOICEwac 8821 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-wrecs 7294  df-recs 7355  df-en 7842  df-card 8648  df-ac 8822 This theorem is referenced by:  axac10  36618
 Copyright terms: Public domain W3C validator