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Mirrors > Home > MPE Home > Th. List > ordon | Structured version Visualization version GIF version |
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 6409 | . 2 ⊢ Tr On | |
2 | epweon 7794 | . 2 ⊢ E We On | |
3 | df-ord 6389 | . 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 5265 E cep 5588 We wwe 5640 Ord word 6385 Oncon0 6386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 |
This theorem is referenced by: onprc 7797 ssorduni 7798 ordeleqon 7801 ordsson 7802 onint 7810 ordsuci 7828 sucexeloniOLD 7830 suceloniOLD 7832 limon 7856 tfi 7874 ordom 7897 ordtypelem2 9557 hartogs 9582 card2on 9592 tskwe 9988 alephsmo 10140 ondomon 10601 dford3lem2 43016 dford3 43017 tfsconcatlem 43326 iunord 48907 |
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