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Mirrors > Home > MPE Home > Th. List > Mathboxes > dford5reg | Structured version Visualization version GIF version |
Description: Given ax-reg 8497, an ordinal is a transitive class totally ordered by epsilon. (Contributed by Scott Fenton, 28-Jan-2011.) |
Ref | Expression |
---|---|
dford5reg | ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ord 5726 | . 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
2 | zfregfr 8509 | . . . 4 ⊢ E Fr 𝐴 | |
3 | df-we 5075 | . . . 4 ⊢ ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ E Or 𝐴)) | |
4 | 2, 3 | mpbiran 953 | . . 3 ⊢ ( E We 𝐴 ↔ E Or 𝐴) |
5 | 4 | anbi2i 730 | . 2 ⊢ ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ E Or 𝐴)) |
6 | 1, 5 | bitri 264 | 1 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 Tr wtr 4752 E cep 5028 Or wor 5034 Fr wfr 5070 We wwe 5072 Ord word 5722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-reg 8497 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-eprel 5029 df-fr 5073 df-we 5075 df-ord 5726 |
This theorem is referenced by: (None) |
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