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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 6851 | . . 3 ⊢ ∪ 𝐴 ∈ V |
3 | numth3 9175 | . . 3 ⊢ (∪ 𝐴 ∈ V → ∪ 𝐴 ∈ dom card) | |
4 | 2, 3 | mp1i 13 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ ∅ → ∪ 𝐴 ∈ dom card) |
5 | neirr 2791 | . . 3 ⊢ ¬ ∅ ≠ ∅ | |
6 | neeq1 2844 | . . . 4 ⊢ (𝑥 = ∅ → (𝑥 ≠ ∅ ↔ ∅ ≠ ∅)) | |
7 | 6 | rspccv 3279 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ ∅ → (∅ ∈ 𝐴 → ∅ ≠ ∅)) |
8 | 5, 7 | mtoi 189 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ≠ ∅ → ¬ ∅ ∈ 𝐴) |
9 | ac5num 8742 | . 2 ⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)) | |
10 | 4, 8, 9 | syl2anc 691 | 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 383 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ∅c0 3874 ∪ cuni 4372 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 cardccrd 8644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-ac2 9168 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-wrecs 7294 df-recs 7355 df-en 7842 df-card 8648 df-ac 8822 |
This theorem is referenced by: acunirnmpt 28841 |
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