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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 6379 | . 2 ⊢ Tr On | |
2 | epweon 7749 | . 2 ⊢ E We On | |
3 | df-ord 6359 | . 2 ⊢ (Ord On ↔ (Tr On ∧ E We On)) | |
4 | 1, 2, 3 | mpbir2an 710 | 1 ⊢ Ord On |
Colors of variables: wff setvar class |
Syntax hints: Tr wtr 5261 E cep 5575 We wwe 5626 Ord word 6355 Oncon0 6356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-tr 5262 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-ord 6359 df-on 6360 |
This theorem is referenced by: onprc 7752 ssorduni 7753 ordeleqon 7756 ordsson 7757 onint 7765 ordsuci 7783 sucexeloniOLD 7785 suceloniOLD 7787 limon 7811 tfi 7829 ordom 7852 ordtypelem2 9501 hartogs 9526 card2on 9536 tskwe 9932 alephsmo 10084 ondomon 10545 dford3lem2 41637 dford3 41638 tfsconcatlem 41957 iunord 47561 |
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