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Mirrors > Home > MPE Home > Th. List > omsson | Structured version Visualization version GIF version |
Description: Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
omsson | ⊢ ω ⊆ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfom2 7562 | . 2 ⊢ ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}} | |
2 | 1 | ssrab3 4008 | 1 ⊢ ω ⊆ On |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 {crab 3110 ⊆ wss 3881 Oncon0 6159 Lim wlim 6160 suc csuc 6161 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 |
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-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: limomss 7565 nnon 7566 ordom 7569 omssnlim 7574 omsinds 7580 nnunifi 8753 unblem1 8754 unblem2 8755 unblem3 8756 unblem4 8757 isfinite2 8760 card2inf 9003 ackbij1lem16 9646 ackbij1lem18 9648 fin23lem26 9736 fin23lem27 9739 isf32lem5 9768 fin1a2lem6 9816 pwfseqlem3 10071 tskinf 10180 grothomex 10240 ltsopi 10299 dmaddpi 10301 dmmulpi 10302 2ndcdisj 22061 finminlem 33779 |
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