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Mirrors > Home > MPE Home > Th. List > winainf | Structured version Visualization version GIF version |
Description: A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.) |
Ref | Expression |
---|---|
winainf | ⊢ (𝐴 ∈ Inaccw → ω ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elwina 10676 | . 2 ⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) | |
2 | cfon 10245 | . . . 4 ⊢ (cf‘𝐴) ∈ On | |
3 | eleq1 2813 | . . . 4 ⊢ ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ On ↔ 𝐴 ∈ On)) | |
4 | 2, 3 | mpbii 232 | . . 3 ⊢ ((cf‘𝐴) = 𝐴 → 𝐴 ∈ On) |
5 | winainflem 10683 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) → ω ⊆ 𝐴) | |
6 | 4, 5 | syl3an2 1161 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) → ω ⊆ 𝐴) |
7 | 1, 6 | sylbi 216 | 1 ⊢ (𝐴 ∈ Inaccw → ω ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ∃wrex 3062 ⊆ wss 3940 ∅c0 4314 class class class wbr 5138 Oncon0 6354 ‘cfv 6533 ωcom 7848 ≺ csdm 8933 cfccf 9927 Inaccwcwina 10672 |
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-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
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-rab 3425 df-v 3468 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-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-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-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-om 7849 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-card 9929 df-cf 9931 df-wina 10674 |
This theorem is referenced by: winalim 10685 winalim2 10686 gchina 10689 inar1 10765 |
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