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Mirrors > Home > MPE Home > Th. List > ac6 | Structured version Visualization version GIF version |
Description: Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a larger set 𝐵, where 𝜑 depends on 𝑥 (the natural number) and 𝑦 (to specify a member of 𝐵). A stronger version of this theorem, ac6s 10098, allows 𝐵 to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.) |
Ref | Expression |
---|---|
ac6.1 | ⊢ 𝐴 ∈ V |
ac6.2 | ⊢ 𝐵 ∈ V |
ac6.3 | ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ac6 | ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ac6.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | ac6.2 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | ssrab2 3993 | . . . . . 6 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵 | |
4 | 3 | rgenw 3073 | . . . . 5 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵 |
5 | iunss 4954 | . . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵) | |
6 | 4, 5 | mpbir 234 | . . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵 |
7 | 2, 6 | ssexi 5215 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V |
8 | numth3 10084 | . . 3 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card) | |
9 | 7, 8 | ax-mp 5 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card |
10 | ac6.3 | . . 3 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) | |
11 | 10 | ac6num 10093 | . 2 ⊢ ((𝐴 ∈ V ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
12 | 1, 9, 11 | mp3an12 1453 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 {crab 3065 Vcvv 3408 ⊆ wss 3866 ∪ ciun 4904 dom cdm 5551 ⟶wf 6376 ‘cfv 6380 cardccrd 9551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-ac2 10077 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-wrecs 8047 df-recs 8108 df-en 8627 df-card 9555 df-ac 9730 |
This theorem is referenced by: ac6c4 10095 ac6s 10098 wlkiswwlksupgr2 27961 |
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