![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7873 | . . 3 ⊢ ∅ ∈ ω | |
2 | 1 | ne0ii 4330 | . 2 ⊢ ω ≠ ∅ |
3 | cfom 10256 | . 2 ⊢ (cf‘ω) = ω | |
4 | nnfi 9164 | . . . . 5 ⊢ (𝑥 ∈ ω → 𝑥 ∈ Fin) | |
5 | pwfi 9175 | . . . . 5 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin) | |
6 | 4, 5 | sylib 217 | . . . 4 ⊢ (𝑥 ∈ ω → 𝒫 𝑥 ∈ Fin) |
7 | isfinite 9644 | . . . 4 ⊢ (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝑥 ≺ ω) | |
8 | 6, 7 | sylib 217 | . . 3 ⊢ (𝑥 ∈ ω → 𝒫 𝑥 ≺ ω) |
9 | 8 | rgen 3055 | . 2 ⊢ ∀𝑥 ∈ ω 𝒫 𝑥 ≺ ω |
10 | elina 10679 | . 2 ⊢ (ω ∈ Inacc ↔ (ω ≠ ∅ ∧ (cf‘ω) = ω ∧ ∀𝑥 ∈ ω 𝒫 𝑥 ≺ ω)) | |
11 | 2, 3, 9, 10 | mpbir3an 1338 | 1 ⊢ ω ∈ Inacc |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ∅c0 4315 𝒫 cpw 4595 class class class wbr 5139 ‘cfv 6534 ωcom 7849 ≺ csdm 8935 Fincfn 8936 cfccf 9929 Inacccina 10675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-cf 9933 df-ina 10677 |
This theorem is referenced by: r1omALT 10768 r1omtsk 10771 |
Copyright terms: Public domain | W3C validator |