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Mirrors > Home > MPE Home > Th. List > Mathboxes > dford5reg | Structured version Visualization version GIF version |
Description: Given ax-reg 9630, 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 6389 | . 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
2 | zfregfr 9643 | . . . 4 ⊢ E Fr 𝐴 | |
3 | df-we 5643 | . . . 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 5265 E cep 5588 Or wor 5596 Fr wfr 5638 We wwe 5640 Ord word 6385 |
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-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-reg 9630 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 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-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-eprel 5589 df-fr 5641 df-we 5643 df-ord 6389 |
This theorem is referenced by: (None) |
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