Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10441 | . 2 ⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) | |
2 | cfon 10010 | . . . 4 ⊢ (cf‘𝐴) ∈ On | |
3 | eleq1 2828 | . . . 4 ⊢ ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ On ↔ 𝐴 ∈ On)) | |
4 | 2, 3 | mpbii 232 | . . 3 ⊢ ((cf‘𝐴) = 𝐴 → 𝐴 ∈ On) |
5 | winainflem 10448 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) → ω ⊆ 𝐴) | |
6 | 4, 5 | syl3an2 1163 | . 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 1086 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∀wral 3066 ∃wrex 3067 ⊆ wss 3892 ∅c0 4262 class class class wbr 5079 Oncon0 6264 ‘cfv 6431 ωcom 7704 ≺ csdm 8713 cfccf 9694 Inaccwcwina 10437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 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 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-om 7705 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-card 9696 df-cf 9698 df-wina 10439 |
This theorem is referenced by: winalim 10450 winalim2 10451 gchina 10454 inar1 10530 |
Copyright terms: Public domain | W3C validator |