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Mirrors > Home > MPE Home > Th. List > ac5b | Structured version Visualization version GIF version |
Description: Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.) |
Ref | Expression |
---|---|
ac5b.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ac5b | ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ ∅ → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ac5b.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | 1 | uniex 7447 | . . 3 ⊢ ∪ 𝐴 ∈ V |
3 | numth3 9881 | . . 3 ⊢ (∪ 𝐴 ∈ V → ∪ 𝐴 ∈ dom card) | |
4 | 2, 3 | mp1i 13 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ ∅ → ∪ 𝐴 ∈ dom card) |
5 | neirr 2996 | . . 3 ⊢ ¬ ∅ ≠ ∅ | |
6 | neeq1 3049 | . . . 4 ⊢ (𝑥 = ∅ → (𝑥 ≠ ∅ ↔ ∅ ≠ ∅)) | |
7 | 6 | rspccv 3568 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ ∅ → (∅ ∈ 𝐴 → ∅ ≠ ∅)) |
8 | 5, 7 | mtoi 202 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ ∅ → ¬ ∅ ∈ 𝐴) |
9 | ac5num 9447 | . 2 ⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)) | |
10 | 4, 8, 9 | syl2anc 587 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ ∅ → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 Vcvv 3441 ∅c0 4243 ∪ cuni 4800 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 cardccrd 9348 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-ac2 9874 |
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 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-wrecs 7930 df-recs 7991 df-en 8493 df-card 9352 df-ac 9527 |
This theorem is referenced by: acunirnmpt 30422 fnpreimac 30434 |
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