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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 us 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 7346 | . . 3 ⊢ ∅ ∈ ω | |
2 | 1 | ne0ii 4153 | . 2 ⊢ ω ≠ ∅ |
3 | cfom 9401 | . 2 ⊢ (cf‘ω) = ω | |
4 | nnfi 8422 | . . . . 5 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin) | |
5 | pwfi 8530 | . . . . 5 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin) | |
6 | 4, 5 | sylib 210 | . . . 4 ⊢ (𝑥 ∈ ω → 𝒫 𝑥 ∈ Fin) |
7 | isfinite 8826 | . . . 4 ⊢ (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝑥 ≺ ω) | |
8 | 6, 7 | sylib 210 | . . 3 ⊢ (𝑥 ∈ ω → 𝒫 𝑥 ≺ ω) |
9 | 8 | rgen 3131 | . 2 ⊢ ∀𝑥 ∈ ω 𝒫 𝑥 ≺ ω |
10 | elina 9824 | . 2 ⊢ (ω ∈ Inacc ↔ (ω ≠ ∅ ∧ (cf‘ω) = ω ∧ ∀𝑥 ∈ ω 𝒫 𝑥 ≺ ω)) | |
11 | 2, 3, 9, 10 | mpbir3an 1447 | 1 ⊢ ω ∈ Inacc |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ∀wral 3117 ∅c0 4144 𝒫 cpw 4378 class class class wbr 4873 ‘cfv 6123 ωcom 7326 ≺ csdm 8221 Fincfn 8222 cfccf 9076 Inacccina 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 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 |
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 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-cf 9080 df-ina 9822 |
This theorem is referenced by: r1omALT 9913 r1omtsk 9916 |
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