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| Mirrors > Home > MPE Home > Th. List > omelon | Structured version Visualization version GIF version | ||
| Description: Omega is an ordinal number. Theorem 1.22 of [Schloeder] p. 3. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
| Ref | Expression |
|---|---|
| omelon | ⊢ ω ∈ On |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | omex 9683 | . 2 ⊢ ω ∈ V | |
| 2 | omelon2 7900 | . 2 ⊢ (ω ∈ V → ω ∈ On) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ ω ∈ On |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 Oncon0 6384 ωcom 7887 |
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-om 7888 |
| This theorem is referenced by: oancom 9691 cnfcomlem 9739 cnfcom 9740 cnfcom2lem 9741 cnfcom2 9742 cnfcom3lem 9743 cnfcom3 9744 cnfcom3clem 9745 cardom 10026 infxpenlem 10053 xpomen 10055 infxpidm2 10057 infxpenc 10058 infxpenc2lem1 10059 infxpenc2 10062 alephon 10109 infenaleph 10131 iunfictbso 10154 dfac12k 10188 infunsdom1 10252 domtriomlem 10482 iunctb 10614 pwcfsdom 10623 canthp1lem2 10693 pwfseqlem4a 10701 pwfseqlem4 10702 pwfseqlem5 10703 wunex3 10781 znnen 16248 qnnen 16249 cygctb 19910 2ndcctbss 23463 2ndcomap 23466 2ndcsep 23467 tx1stc 23658 tx2ndc 23659 met1stc 24534 met2ndci 24535 re2ndc 24822 uniiccdif 25613 dyadmbl 25635 opnmblALT 25638 mbfimaopnlem 25690 aannenlem3 26372 n0ssold 28355 exrecfnlem 37380 poimirlem32 37659 numinfctb 43115 onexomgt 43253 onexlimgt 43255 onexoegt 43256 1oaomeqom 43306 oaabsb 43307 oaordnrex 43308 oaordnr 43309 2omomeqom 43316 omnord1ex 43317 omnord1 43318 nnoeomeqom 43325 oenord1 43329 oaomoencom 43330 cantnftermord 43333 cantnfub 43334 cantnf2 43338 nnawordexg 43340 dflim5 43342 oacl2g 43343 onmcl 43344 omabs2 43345 omcl2 43346 tfsnfin 43365 ofoaf 43368 ofoafo 43369 naddcnff 43375 naddcnffo 43377 naddcnfcom 43379 naddcnfid1 43380 naddcnfid2 43381 naddcnfass 43382 naddwordnexlem0 43409 naddwordnexlem1 43410 naddwordnexlem3 43412 oawordex3 43413 naddwordnexlem4 43414 infordmin 43545 minregex 43547 omiscard 43556 sucomisnotcard 43557 aleph1min 43570 alephiso3 43572 wfaxinf2 45018 |
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