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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dford5reg | Structured version Visualization version GIF version | ||
| Description: Given ax-reg 9632, an ordinal is a transitive class totally ordered by the membership relation. (Contributed by Scott Fenton, 28-Jan-2011.) |
| Ref | Expression |
|---|---|
| dford5reg | ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ord 6387 | . 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
| 2 | zfregfr 9645 | . . . 4 ⊢ E Fr 𝐴 | |
| 3 | df-we 5639 | . . . 4 ⊢ ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ E Or 𝐴)) | |
| 4 | 2, 3 | mpbiran 709 | . . 3 ⊢ ( E We 𝐴 ↔ E Or 𝐴) |
| 5 | 4 | anbi2i 623 | . 2 ⊢ ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ E Or 𝐴)) |
| 6 | 1, 5 | bitri 275 | 1 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 Tr wtr 5259 E cep 5583 Or wor 5591 Fr wfr 5634 We wwe 5636 Ord word 6383 |
| 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-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-reg 9632 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 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-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-fr 5637 df-we 5639 df-ord 6387 |
| This theorem is referenced by: (None) |
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