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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 10290 | . 2 ⊢ (cf‘ω) ⊆ ω | |
| 2 | limom 7899 | . . . 4 ⊢ Lim ω | |
| 3 | omex 9679 | . . . . 5 ⊢ ω ∈ V | |
| 4 | 3 | cflim2 10299 | . . . 4 ⊢ (Lim ω ↔ Lim (cf‘ω)) |
| 5 | 2, 4 | mpbi 230 | . . 3 ⊢ Lim (cf‘ω) |
| 6 | limomss 7888 | . . 3 ⊢ (Lim (cf‘ω) → ω ⊆ (cf‘ω)) | |
| 7 | 5, 6 | ax-mp 5 | . 2 ⊢ ω ⊆ (cf‘ω) |
| 8 | 1, 7 | eqssi 3999 | 1 ⊢ (cf‘ω) = ω |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ⊆ wss 3950 Lim wlim 6383 ‘cfv 6559 ωcom 7883 cfccf 9973 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-inf2 9677 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-se 5636 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 df-riota 7386 df-ov 7432 df-om 7884 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-card 9975 df-cf 9977 |
| This theorem is referenced by: pwcfsdom 10619 alephom 10621 omina 10727 |
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