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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 9824 | . 2 ⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) | |
2 | cfon 9393 | . . . 4 ⊢ (cf‘𝐴) ∈ On | |
3 | eleq1 2895 | . . . 4 ⊢ ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ On ↔ 𝐴 ∈ On)) | |
4 | 2, 3 | mpbii 225 | . . 3 ⊢ ((cf‘𝐴) = 𝐴 → 𝐴 ∈ On) |
5 | winainflem 9831 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) → ω ⊆ 𝐴) | |
6 | 4, 5 | syl3an2 1209 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) → ω ⊆ 𝐴) |
7 | 1, 6 | sylbi 209 | 1 ⊢ (𝐴 ∈ Inaccw → ω ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 ∀wral 3118 ∃wrex 3119 ⊆ wss 3799 ∅c0 4145 class class class wbr 4874 Oncon0 5964 ‘cfv 6124 ωcom 7327 ≺ csdm 8222 cfccf 9077 Inaccwcwina 9820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-om 7328 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-card 9079 df-cf 9081 df-wina 9822 |
This theorem is referenced by: winalim 9833 winalim2 9834 gchina 9837 inar1 9913 |
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