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| 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 7896 | . 2 ⊢ Tr ω | |
| 2 | omsson 7891 | . 2 ⊢ ω ⊆ On | |
| 3 | ordon 7797 | . 2 ⊢ Ord On | |
| 4 | trssord 6401 | . 2 ⊢ ((Tr ω ∧ ω ⊆ On ∧ Ord On) → Ord ω) | |
| 5 | 1, 2, 3, 4 | mp3an 1463 | 1 ⊢ Ord ω | 
| Colors of variables: wff setvar class | 
| Syntax hints: ⊆ wss 3951 Tr wtr 5259 Ord word 6383 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 | 
| 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-om 7888 | 
| This theorem is referenced by: omon 7899 limom 7903 ssnlim 7907 peano5 7915 omsucelsucb 8498 nnarcl 8654 nnawordex 8675 oaabslem 8685 oaabs2 8687 omabslem 8688 onomeneqOLD 9266 ominf 9294 ominfOLD 9295 findcard3 9318 findcard3OLD 9319 nnsdomg 9335 nnsdomgOLD 9336 dffi3 9471 wofib 9585 alephgeom 10122 iscard3 10133 iunfictbso 10154 unctb 10244 ackbij2lem1 10258 ackbij1lem3 10261 ackbij1lem18 10276 ackbij2 10282 cflim2 10303 fin23lem26 10365 fin23lem23 10366 fin23lem27 10368 fin67 10435 alephexp1 10619 pwfseqlem3 10700 pwdjundom 10707 winainflem 10733 wunex2 10778 om2uzoi 13996 ltweuz 14002 fz1isolem 14500 1stcrestlem 23460 om2noseqoi 28309 satfn 35360 hfuni 36185 hfninf 36187 finxpreclem4 37395 oaordnrex 43308 omnord1ex 43317 oenord1ex 43328 omabs2 43345 tfsconcat0b 43359 rn1st 45280 | 
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