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Mirrors > Home > MPE Home > Th. List > winalim | Structured version Visualization version GIF version |
Description: A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.) |
Ref | Expression |
---|---|
winalim | ⊢ (𝐴 ∈ Inaccw → Lim 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | winainf 10118 | . 2 ⊢ (𝐴 ∈ Inaccw → ω ⊆ 𝐴) | |
2 | winacard 10116 | . . 3 ⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴) | |
3 | cardlim 9403 | . . . 4 ⊢ (ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴)) | |
4 | sseq2 3995 | . . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (ω ⊆ (card‘𝐴) ↔ ω ⊆ 𝐴)) | |
5 | limeq 6205 | . . . . 5 ⊢ ((card‘𝐴) = 𝐴 → (Lim (card‘𝐴) ↔ Lim 𝐴)) | |
6 | 4, 5 | bibi12d 348 | . . . 4 ⊢ ((card‘𝐴) = 𝐴 → ((ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴)) ↔ (ω ⊆ 𝐴 ↔ Lim 𝐴))) |
7 | 3, 6 | mpbii 235 | . . 3 ⊢ ((card‘𝐴) = 𝐴 → (ω ⊆ 𝐴 ↔ Lim 𝐴)) |
8 | 2, 7 | syl 17 | . 2 ⊢ (𝐴 ∈ Inaccw → (ω ⊆ 𝐴 ↔ Lim 𝐴)) |
9 | 1, 8 | mpbid 234 | 1 ⊢ (𝐴 ∈ Inaccw → Lim 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 Lim wlim 6194 ‘cfv 6357 ωcom 7582 cardccrd 9366 Inaccwcwina 10106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-1o 8104 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-card 9370 df-cf 9372 df-wina 10108 |
This theorem is referenced by: inar1 10199 inatsk 10202 tskuni 10207 grur1a 10243 |
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