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Mirrors > Home > MPE Home > Th. List > Mathboxes > dford5reg | Structured version Visualization version GIF version |
Description: Given ax-reg 9044, 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 6187 | . 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
2 | zfregfr 9056 | . . . 4 ⊢ E Fr 𝐴 | |
3 | df-we 5509 | . . . 4 ⊢ ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ E Or 𝐴)) | |
4 | 2, 3 | mpbiran 705 | . . 3 ⊢ ( E We 𝐴 ↔ E Or 𝐴) |
5 | 4 | anbi2i 622 | . 2 ⊢ ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ E Or 𝐴)) |
6 | 1, 5 | bitri 276 | 1 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 Tr wtr 5163 E cep 5457 Or wor 5466 Fr wfr 5504 We wwe 5506 Ord word 6183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-reg 9044 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-eprel 5458 df-fr 5507 df-we 5509 df-ord 6187 |
This theorem is referenced by: (None) |
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