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Mirrors > Home > MPE Home > Th. List > ordom | Structured version Visualization version GIF version |
Description: The class of finite ordinals ω is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. Theorem 1.22 of [Schloeder] p. 3. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
ordom | ⊢ Ord ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trom 7885 | . 2 ⊢ Tr ω | |
2 | omsson 7880 | . 2 ⊢ ω ⊆ On | |
3 | ordon 7785 | . 2 ⊢ Ord On | |
4 | trssord 6393 | . 2 ⊢ ((Tr ω ∧ ω ⊆ On ∧ Ord On) → Ord ω) | |
5 | 1, 2, 3, 4 | mp3an 1458 | 1 ⊢ Ord ω |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3947 Tr wtr 5270 Ord word 6375 Oncon0 6376 ωcom 7876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-tr 5271 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-ord 6379 df-on 6380 df-lim 6381 df-om 7877 |
This theorem is referenced by: omon 7888 limom 7892 ssnlim 7896 omsindsOLD 7898 peano5 7905 peano5OLD 7906 omsucelsucb 8488 nnarcl 8646 nnawordex 8667 oaabslem 8677 oaabs2 8679 omabslem 8680 onomeneqOLD 9263 ominf 9292 ominfOLD 9293 findcard3 9319 findcard3OLD 9320 nnsdomg 9336 nnsdomgOLD 9337 dffi3 9474 wofib 9588 alephgeom 10125 iscard3 10136 iunfictbso 10157 unctb 10248 ackbij2lem1 10262 ackbij1lem3 10265 ackbij1lem18 10280 ackbij2 10286 cflim2 10306 fin23lem26 10368 fin23lem23 10369 fin23lem27 10371 fin67 10438 alephexp1 10622 pwfseqlem3 10703 pwdjundom 10710 winainflem 10736 wunex2 10781 om2uzoi 13975 ltweuz 13981 fz1isolem 14480 1stcrestlem 23447 om2noseqoi 28277 satfn 35183 hfuni 36008 hfninf 36010 finxpreclem4 37101 oaordnrex 42961 omnord1ex 42970 oenord1ex 42981 omabs2 42998 tfsconcat0b 43012 rn1st 44883 |
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