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Mirrors > Home > MPE Home > Th. List > ordeq | Structured version Visualization version GIF version |
Description: Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.) |
Ref | Expression |
---|---|
ordeq | ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | treq 5277 | . . 3 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵)) | |
2 | weeq2 5671 | . . 3 ⊢ (𝐴 = 𝐵 → ( E We 𝐴 ↔ E We 𝐵)) | |
3 | 1, 2 | anbi12d 630 | . 2 ⊢ (𝐴 = 𝐵 → ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐵 ∧ E We 𝐵))) |
4 | df-ord 6377 | . 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
5 | df-ord 6377 | . 2 ⊢ (Ord 𝐵 ↔ (Tr 𝐵 ∧ E We 𝐵)) | |
6 | 3, 4, 5 | 3bitr4g 313 | 1 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 Tr wtr 5269 E cep 5585 We wwe 5636 Ord word 6373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-v 3475 df-in 3956 df-ss 3966 df-uni 4913 df-tr 5270 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-ord 6377 |
This theorem is referenced by: elong 6382 limeq 6386 ordelord 6396 ordun 6478 ordeleqon 7790 ordsuc 7822 ordsucOLD 7823 ordzsl 7855 issmo 8375 issmo2 8376 smoeq 8377 smores 8379 smores2 8381 smodm2 8382 smoiso 8389 tfrlem8 8411 ord3 8510 ordtypelem5 9553 ordtypelem7 9555 oicl 9560 oieu 9570 dfsucon 42984 |
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