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Mirrors > Home > MPE Home > Th. List > cfom | Structured version Visualization version GIF version |
Description: Value of the cofinality function at omega (the set of natural numbers). Exercise 4 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Proof shortened by Mario Carneiro, 11-Jun-2015.) |
Ref | Expression |
---|---|
cfom | ⊢ (cf‘ω) = ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfle 10236 | . 2 ⊢ (cf‘ω) ⊆ ω | |
2 | limom 7858 | . . . 4 ⊢ Lim ω | |
3 | omex 9625 | . . . . 5 ⊢ ω ∈ V | |
4 | 3 | cflim2 10245 | . . . 4 ⊢ (Lim ω ↔ Lim (cf‘ω)) |
5 | 2, 4 | mpbi 229 | . . 3 ⊢ Lim (cf‘ω) |
6 | limomss 7847 | . . 3 ⊢ (Lim (cf‘ω) → ω ⊆ (cf‘ω)) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ ω ⊆ (cf‘ω) |
8 | 1, 7 | eqssi 3996 | 1 ⊢ (cf‘ω) = ω |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⊆ wss 3946 Lim wlim 6357 ‘cfv 6535 ωcom 7842 cfccf 9919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-inf2 9623 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-iin 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 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 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7352 df-ov 7399 df-om 7843 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-card 9921 df-cf 9923 |
This theorem is referenced by: pwcfsdom 10565 alephom 10567 omina 10673 |
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