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| Description: The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.) | 
| Ref | Expression | 
|---|---|
| ordon | ⊢ Ord On | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tron 6406 | . 2 ⊢ Tr On | |
| 2 | epweon 7796 | . 2 ⊢ E We On | |
| 3 | df-ord 6386 | . 2 ⊢ (Ord On ↔ (Tr On ∧ E We On)) | |
| 4 | 1, 2, 3 | mpbir2an 711 | 1 ⊢ Ord On | 
| Colors of variables: wff setvar class | 
| Syntax hints: Tr wtr 5258 E cep 5582 We wwe 5635 Ord word 6382 Oncon0 6383 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 | 
| This theorem is referenced by: onprc 7799 ssorduni 7800 ordeleqon 7803 ordsson 7804 onint 7811 ordsuci 7829 sucexeloniOLD 7831 suceloniOLD 7833 limon 7857 tfi 7875 ordom 7898 ordtypelem2 9560 hartogs 9585 card2on 9595 tskwe 9991 alephsmo 10143 ondomon 10604 dford3lem2 43044 dford3 43045 tfsconcatlem 43354 iunord 49250 | 
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