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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 9142 | . . 3 ⊢ (𝑥 ∈ dom card ↔ ∃𝑦 𝑦 We 𝑥) | |
2 | 1 | albii 1915 | . 2 ⊢ (∀𝑥 𝑥 ∈ dom card ↔ ∀𝑥∃𝑦 𝑦 We 𝑥) |
3 | eqv 3389 | . 2 ⊢ (dom card = V ↔ ∀𝑥 𝑥 ∈ dom card) | |
4 | dfac8 9243 | . 2 ⊢ (CHOICE ↔ ∀𝑥∃𝑦 𝑦 We 𝑥) | |
5 | 2, 3, 4 | 3bitr4ri 296 | 1 ⊢ (CHOICE ↔ dom card = V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∀wal 1651 = wceq 1653 ∃wex 1875 ∈ wcel 2157 Vcvv 3383 We wwe 5268 dom cdm 5310 cardccrd 9045 CHOICEwac 9222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-se 5270 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-isom 6108 df-riota 6837 df-wrecs 7643 df-recs 7705 df-en 8194 df-card 9049 df-ac 9223 |
This theorem is referenced by: dfac10c 9246 acacni 9248 dfac12a 9256 dfacfin7 9507 cardeqv 9577 gch2 9783 gchac 9789 lbsexg 19484 acufl 22046 ttac 38376 dfac21 38409 dfacbasgrp 38451 |
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