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Mirrors > Home > MPE Home > Th. List > inawina | Structured version Visualization version GIF version |
Description: Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.) |
Ref | Expression |
---|---|
inawina | ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfon 10279 | . . . . 5 ⊢ (cf‘𝐴) ∈ On | |
2 | eleq1 2817 | . . . . 5 ⊢ ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ On ↔ 𝐴 ∈ On)) | |
3 | 1, 2 | mpbii 232 | . . . 4 ⊢ ((cf‘𝐴) = 𝐴 → 𝐴 ∈ On) |
4 | 3 | 3ad2ant2 1132 | . . 3 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴) → 𝐴 ∈ On) |
5 | idd 24 | . . . 4 ⊢ (𝐴 ∈ On → (𝐴 ≠ ∅ → 𝐴 ≠ ∅)) | |
6 | idd 24 | . . . 4 ⊢ (𝐴 ∈ On → ((cf‘𝐴) = 𝐴 → (cf‘𝐴) = 𝐴)) | |
7 | inawinalem 10713 | . . . 4 ⊢ (𝐴 ∈ On → (∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) | |
8 | 5, 6, 7 | 3anim123d 1440 | . . 3 ⊢ (𝐴 ∈ On → ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴) → (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))) |
9 | 4, 8 | mpcom 38 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴) → (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) |
10 | elina 10711 | . 2 ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴)) | |
11 | elwina 10710 | . 2 ⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) | |
12 | 9, 10, 11 | 3imtr4i 292 | 1 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ∅c0 4323 𝒫 cpw 4603 class class class wbr 5148 Oncon0 6369 ‘cfv 6548 ≺ csdm 8963 cfccf 9961 Inaccwcwina 10706 Inacccina 10707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-card 9963 df-cf 9965 df-wina 10708 df-ina 10709 |
This theorem is referenced by: gchina 10723 inar1 10799 inatsk 10802 tskuni 10807 grur1a 10843 grur1 10844 inaprc 10860 inaex 43734 gruex 43735 |
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