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Mirrors > Home > MPE Home > Th. List > ac6s5 | Structured version Visualization version GIF version |
Description: Generalization of the Axiom of Choice to proper classes. 𝐵 is a collection 𝐵(𝑥) of nonempty, possible proper classes. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by NM, 27-Mar-2006.) |
Ref | Expression |
---|---|
ac6s4.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ac6s5 | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ac6s4.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | ac6s4 10288 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) |
3 | exsimpr 1870 | . 2 ⊢ (∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) | |
4 | 2, 3 | syl 17 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∃wex 1779 ∈ wcel 2104 ≠ wne 2941 ∀wral 3062 Vcvv 3437 ∅c0 4262 Fn wfn 6449 ‘cfv 6454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-reg 9391 ax-inf2 9439 ax-ac2 10261 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-se 5552 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-isom 6463 df-riota 7260 df-ov 7306 df-om 7741 df-2nd 7860 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-en 8761 df-r1 9562 df-rank 9563 df-card 9737 df-ac 9914 |
This theorem is referenced by: (None) |
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