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Mirrors > Home > MPE Home > Th. List > winafp | Structured version Visualization version GIF version |
Description: A nontrivial weakly inaccessible cardinal is a fixed point of the aleph function. (Contributed by Mario Carneiro, 29-May-2014.) |
Ref | Expression |
---|---|
winafp | ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → (ℵ‘𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | winalim2 10687 | . 2 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)) | |
2 | vex 3470 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
3 | limelon 6418 | . . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On) | |
4 | 2, 3 | mpan 687 | . . . . . . . 8 ⊢ (Lim 𝑥 → 𝑥 ∈ On) |
5 | alephle 10079 | . . . . . . . 8 ⊢ (𝑥 ∈ On → 𝑥 ⊆ (ℵ‘𝑥)) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (Lim 𝑥 → 𝑥 ⊆ (ℵ‘𝑥)) |
7 | 6 | ad2antll 726 | . . . . . 6 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)) → 𝑥 ⊆ (ℵ‘𝑥)) |
8 | simprl 768 | . . . . . 6 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)) → (ℵ‘𝑥) = 𝐴) | |
9 | 7, 8 | sseqtrd 4014 | . . . . 5 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)) → 𝑥 ⊆ 𝐴) |
10 | 8 | fveq2d 6885 | . . . . . . . 8 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)) → (cf‘(ℵ‘𝑥)) = (cf‘𝐴)) |
11 | alephsing 10267 | . . . . . . . . 9 ⊢ (Lim 𝑥 → (cf‘(ℵ‘𝑥)) = (cf‘𝑥)) | |
12 | 11 | ad2antll 726 | . . . . . . . 8 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)) → (cf‘(ℵ‘𝑥)) = (cf‘𝑥)) |
13 | 10, 12 | eqtr3d 2766 | . . . . . . 7 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)) → (cf‘𝐴) = (cf‘𝑥)) |
14 | elwina 10677 | . . . . . . . . 9 ⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐴 𝑦 ≺ 𝑧)) | |
15 | 14 | simp2bi 1143 | . . . . . . . 8 ⊢ (𝐴 ∈ Inaccw → (cf‘𝐴) = 𝐴) |
16 | 15 | ad2antrr 723 | . . . . . . 7 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)) → (cf‘𝐴) = 𝐴) |
17 | 13, 16 | eqtr3d 2766 | . . . . . 6 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)) → (cf‘𝑥) = 𝐴) |
18 | cfle 10245 | . . . . . 6 ⊢ (cf‘𝑥) ⊆ 𝑥 | |
19 | 17, 18 | eqsstrrdi 4029 | . . . . 5 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)) → 𝐴 ⊆ 𝑥) |
20 | 9, 19 | eqssd 3991 | . . . 4 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)) → 𝑥 = 𝐴) |
21 | 20 | fveq2d 6885 | . . 3 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)) → (ℵ‘𝑥) = (ℵ‘𝐴)) |
22 | 21, 8 | eqtr3d 2766 | . 2 ⊢ (((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) ∧ ((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)) → (ℵ‘𝐴) = 𝐴) |
23 | 1, 22 | exlimddv 1930 | 1 ⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → (ℵ‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ∃wrex 3062 Vcvv 3466 ⊆ wss 3940 ∅c0 4314 class class class wbr 5138 Oncon0 6354 Lim wlim 6355 ‘cfv 6533 ωcom 7848 ≺ csdm 8934 ℵcale 9927 cfccf 9928 Inaccwcwina 10673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-smo 8341 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-oi 9501 df-har 9548 df-card 9930 df-aleph 9931 df-cf 9932 df-acn 9933 df-wina 10675 |
This theorem is referenced by: winafpi 10689 |
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