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Mirrors > Home > MPE Home > Th. List > dfac10 | Structured version Visualization version GIF version |
Description: Axiom of Choice equivalent: the cardinality function measures every set. (Contributed by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
dfac10 | ⊢ (CHOICE ↔ dom card = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ween 9059 | . . 3 ⊢ (𝑥 ∈ dom card ↔ ∃𝑦 𝑦 We 𝑥) | |
2 | 1 | albii 1895 | . 2 ⊢ (∀𝑥 𝑥 ∈ dom card ↔ ∀𝑥∃𝑦 𝑦 We 𝑥) |
3 | eqv 3356 | . 2 ⊢ (dom card = V ↔ ∀𝑥 𝑥 ∈ dom card) | |
4 | dfac8 9160 | . 2 ⊢ (CHOICE ↔ ∀𝑥∃𝑦 𝑦 We 𝑥) | |
5 | 2, 3, 4 | 3bitr4ri 293 | 1 ⊢ (CHOICE ↔ dom card = V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∀wal 1629 = wceq 1631 ∃wex 1852 ∈ wcel 2145 Vcvv 3351 We wwe 5208 dom cdm 5250 cardccrd 8962 CHOICEwac 9139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-isom 6041 df-riota 6755 df-wrecs 7560 df-recs 7622 df-en 8111 df-card 8966 df-ac 9140 |
This theorem is referenced by: dfac10c 9163 acacni 9165 dfac12a 9173 dfacfin7 9424 cardeqv 9494 gch2 9700 gchac 9706 lbsexg 19380 acufl 21942 ttac 38130 dfac21 38163 dfacbasgrp 38205 |
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