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Mirrors > Home > MPE Home > Th. List > omina | Structured version Visualization version GIF version |
Description: ω is a strongly inaccessible cardinal. (Many definitions of "inaccessible" explicitly disallow ω as an inaccessible cardinal, but this choice allows to reuse our results for inaccessibles for ω.) (Contributed by Mario Carneiro, 29-May-2014.) |
Ref | Expression |
---|---|
omina | ⊢ ω ∈ Inacc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 7888 | . . 3 ⊢ ∅ ∈ ω | |
2 | 1 | ne0ii 4333 | . 2 ⊢ ω ≠ ∅ |
3 | cfom 10281 | . 2 ⊢ (cf‘ω) = ω | |
4 | nnfi 9185 | . . . . 5 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin) | |
5 | pwfi 9196 | . . . . 5 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin) | |
6 | 4, 5 | sylib 217 | . . . 4 ⊢ (𝑥 ∈ ω → 𝒫 𝑥 ∈ Fin) |
7 | isfinite 9669 | . . . 4 ⊢ (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝑥 ≺ ω) | |
8 | 6, 7 | sylib 217 | . . 3 ⊢ (𝑥 ∈ ω → 𝒫 𝑥 ≺ ω) |
9 | 8 | rgen 3059 | . 2 ⊢ ∀𝑥 ∈ ω 𝒫 𝑥 ≺ ω |
10 | elina 10704 | . 2 ⊢ (ω ∈ Inacc ↔ (ω ≠ ∅ ∧ (cf‘ω) = ω ∧ ∀𝑥 ∈ ω 𝒫 𝑥 ≺ ω)) | |
11 | 2, 3, 9, 10 | mpbir3an 1339 | 1 ⊢ ω ∈ Inacc |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∀wral 3057 ∅c0 4318 𝒫 cpw 4598 class class class wbr 5142 ‘cfv 6542 ωcom 7864 ≺ csdm 8956 Fincfn 8957 cfccf 9954 Inacccina 10700 |
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 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 |
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 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-card 9956 df-cf 9958 df-ina 10702 |
This theorem is referenced by: r1omALT 10793 r1omtsk 10796 |
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