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Mirrors > Home > MPE Home > Th. List > omelon | Structured version Visualization version GIF version |
Description: Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
Ref | Expression |
---|---|
omelon | ⊢ ω ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omex 9090 | . 2 ⊢ ω ∈ V | |
2 | omelon2 7572 | . 2 ⊢ (ω ∈ V → ω ∈ On) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ω ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 Vcvv 3441 Oncon0 6159 ωcom 7560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-om 7561 |
This theorem is referenced by: oancom 9098 cnfcomlem 9146 cnfcom 9147 cnfcom2lem 9148 cnfcom2 9149 cnfcom3lem 9150 cnfcom3 9151 cnfcom3clem 9152 cardom 9399 infxpenlem 9424 xpomen 9426 infxpidm2 9428 infxpenc 9429 infxpenc2lem1 9430 infxpenc2 9433 alephon 9480 infenaleph 9502 iunfictbso 9525 dfac12k 9558 infunsdom1 9624 domtriomlem 9853 iunctb 9985 pwcfsdom 9994 canthp1lem2 10064 pwfseqlem4a 10072 pwfseqlem4 10073 pwfseqlem5 10074 wunex3 10152 znnen 15557 qnnen 15558 cygctb 19005 2ndcctbss 22060 2ndcomap 22063 2ndcsep 22064 tx1stc 22255 tx2ndc 22256 met1stc 23128 met2ndci 23129 re2ndc 23406 uniiccdif 24182 dyadmbl 24204 opnmblALT 24207 mbfimaopnlem 24259 aannenlem3 24926 exrecfnlem 34796 poimirlem32 35089 numinfctb 40047 infordmin 40240 aleph1min 40256 alephiso3 40258 |
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