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Mirrors > Home > MPE Home > Th. List > Mathboxes > dford5reg | Structured version Visualization version GIF version |
Description: Given ax-reg 9056, 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 6194 | . 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
2 | zfregfr 9068 | . . . 4 ⊢ E Fr 𝐴 | |
3 | df-we 5516 | . . . 4 ⊢ ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ E Or 𝐴)) | |
4 | 2, 3 | mpbiran 707 | . . 3 ⊢ ( E We 𝐴 ↔ E Or 𝐴) |
5 | 4 | anbi2i 624 | . 2 ⊢ ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ E Or 𝐴)) |
6 | 1, 5 | bitri 277 | 1 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 Tr wtr 5172 E cep 5464 Or wor 5473 Fr wfr 5511 We wwe 5513 Ord word 6190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-reg 9056 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-eprel 5465 df-fr 5514 df-we 5516 df-ord 6194 |
This theorem is referenced by: (None) |
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