Theorem dfac10b 8961

Description: Axiom of Choice equivalent: every set is equinumerous to an ordinal (quantifier-free short cryptic version alluded to in df-ac 8939). (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.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 ⊢ 𝑥 ∈ V
2	1	elima 5471	. . . 4 ⊢ (𝑥 ∈ ( ≈ “ On) ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑥)
3	2	bicomi 214	. . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝑥 ↔ 𝑥 ∈ ( ≈ “ On))
4	3	albii 1747	. 2 ⊢ (∀𝑥∃𝑦 ∈ On 𝑦 ≈ 𝑥 ↔ ∀𝑥 𝑥 ∈ ( ≈ “ On))
5		dfac10c 8960	. 2 ⊢ (CHOICE ↔ ∀𝑥∃𝑦 ∈ On 𝑦 ≈ 𝑥)
6		eqv 3205	. 2 ⊢ (( ≈ “ On) = V ↔ ∀𝑥 𝑥 ∈ ( ≈ “ On))
7	4, 5, 6	3bitr4i 292	1 ⊢ (CHOICE ↔ ( ≈ “ On) = V)

Syntax hints:   ↔ wb 196  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200   class class class wbr 4653   “ cima 5117  Oncon0 5723   ≈ cen 7952  CHOICEwac 8938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-wrecs 7407  df-recs 7468  df-en 7956  df-card 8765  df-ac 8939

This theorem is referenced by:  axac10  37600