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| Mirrors > Home > MPE Home > Th. List > axdc2 | Structured version Visualization version GIF version | ||
| Description: An apparent strengthening of ax-dc 10486 (but derived from it) which shows that there is a denumerable sequence 𝑔 for any function that maps elements of a set 𝐴 to nonempty subsets of 𝐴 such that 𝑔(𝑥 + 1) ∈ 𝐹(𝑔(𝑥)) for all 𝑥 ∈ ω. The finitistic version of this can be proven by induction, but the infinite version requires this new axiom. (Contributed by Mario Carneiro, 25-Jan-2013.) |
| Ref | Expression |
|---|---|
| axdc2.1 | ⊢ 𝐴 ∈ V |
| Ref | Expression |
|---|---|
| axdc2 | ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | axdc2.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | eleq1w 2824 | . . . . 5 ⊢ (𝑠 = 𝑥 → (𝑠 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
| 3 | 2 | adantr 480 | . . . 4 ⊢ ((𝑠 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑠 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) |
| 4 | fveq2 6906 | . . . . . 6 ⊢ (𝑠 = 𝑥 → (𝐹‘𝑠) = (𝐹‘𝑥)) | |
| 5 | 4 | eleq2d 2827 | . . . . 5 ⊢ (𝑠 = 𝑥 → (𝑡 ∈ (𝐹‘𝑠) ↔ 𝑡 ∈ (𝐹‘𝑥))) |
| 6 | eleq1w 2824 | . . . . 5 ⊢ (𝑡 = 𝑦 → (𝑡 ∈ (𝐹‘𝑥) ↔ 𝑦 ∈ (𝐹‘𝑥))) | |
| 7 | 5, 6 | sylan9bb 509 | . . . 4 ⊢ ((𝑠 = 𝑥 ∧ 𝑡 = 𝑦) → (𝑡 ∈ (𝐹‘𝑠) ↔ 𝑦 ∈ (𝐹‘𝑥))) |
| 8 | 3, 7 | anbi12d 632 | . . 3 ⊢ ((𝑠 = 𝑥 ∧ 𝑡 = 𝑦) → ((𝑠 ∈ 𝐴 ∧ 𝑡 ∈ (𝐹‘𝑠)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥)))) |
| 9 | 8 | cbvopabv 5216 | . 2 ⊢ {〈𝑠, 𝑡〉 ∣ (𝑠 ∈ 𝐴 ∧ 𝑡 ∈ (𝐹‘𝑠))} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))} |
| 10 | fveq2 6906 | . . 3 ⊢ (𝑛 = 𝑥 → (ℎ‘𝑛) = (ℎ‘𝑥)) | |
| 11 | 10 | cbvmptv 5255 | . 2 ⊢ (𝑛 ∈ ω ↦ (ℎ‘𝑛)) = (𝑥 ∈ ω ↦ (ℎ‘𝑥)) |
| 12 | 1, 9, 11 | axdc2lem 10488 | 1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔‘𝑘)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 Vcvv 3480 ∖ cdif 3948 ∅c0 4333 𝒫 cpw 4600 {csn 4626 {copab 5205 ↦ cmpt 5225 suc csuc 6386 ⟶wf 6557 ‘cfv 6561 ωcom 7887 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-dc 10486 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-om 7888 df-1o 8506 |
| This theorem is referenced by: axdc3lem4 10493 |
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