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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 6214 | . 2 ⊢ Tr On | |
2 | epweon 7538 | . 2 ⊢ E We On | |
3 | df-ord 6194 | . 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 5146 E cep 5444 We wwe 5493 Ord word 6190 Oncon0 6191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-tr 5147 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-ord 6194 df-on 6195 |
This theorem is referenced by: onprc 7540 ssorduni 7541 ordeleqon 7544 ordsson 7545 onint 7552 suceloni 7570 limon 7593 tfi 7610 ordom 7632 ordtypelem2 9113 hartogs 9138 card2on 9148 tskwe 9531 alephsmo 9681 ondomon 10142 dford3lem2 40493 dford3 40494 iunord 45996 |
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