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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 7740 | . . 3 ⊢ ∪ 𝐴 ∈ V |
3 | numth3 10487 | . . 3 ⊢ (∪ 𝐴 ∈ V → ∪ 𝐴 ∈ dom card) | |
4 | 2, 3 | mp1i 13 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ ∅ → ∪ 𝐴 ∈ dom card) |
5 | neirr 2945 | . . 3 ⊢ ¬ ∅ ≠ ∅ | |
6 | neeq1 2999 | . . . 4 ⊢ (𝑥 = ∅ → (𝑥 ≠ ∅ ↔ ∅ ≠ ∅)) | |
7 | 6 | rspccv 3605 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ ∅ → (∅ ∈ 𝐴 → ∅ ≠ ∅)) |
8 | 5, 7 | mtoi 198 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ ∅ → ¬ ∅ ∈ 𝐴) |
9 | ac5num 10053 | . 2 ⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)) | |
10 | 4, 8, 9 | syl2anc 583 | 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 395 ∃wex 1774 ∈ wcel 2099 ≠ wne 2936 ∀wral 3057 Vcvv 3470 ∅c0 4318 ∪ cuni 4903 dom cdm 5672 ⟶wf 6538 ‘cfv 6542 cardccrd 9952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-ac2 10480 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-en 8958 df-card 9956 df-ac 10133 |
This theorem is referenced by: acunirnmpt 32438 fnpreimac 32450 |
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