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Mirrors > Home > ILE Home > Th. List > ordeq | 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 4002 | . . 3 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵)) | |
2 | raleq 2603 | . . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 Tr 𝑥 ↔ ∀𝑥 ∈ 𝐵 Tr 𝑥)) | |
3 | 1, 2 | anbi12d 464 | . 2 ⊢ (𝐴 = 𝐵 → ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥) ↔ (Tr 𝐵 ∧ ∀𝑥 ∈ 𝐵 Tr 𝑥))) |
4 | dford3 4259 | . 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥)) | |
5 | dford3 4259 | . 2 ⊢ (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥 ∈ 𝐵 Tr 𝑥)) | |
6 | 3, 4, 5 | 3bitr4g 222 | 1 ⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1316 ∀wral 2393 Tr wtr 3996 Ord word 4254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-in 3047 df-ss 3054 df-uni 3707 df-tr 3997 df-iord 4258 |
This theorem is referenced by: elong 4265 limeq 4269 ordelord 4273 ordtriexmidlem 4405 2ordpr 4409 issmo 6153 issmo2 6154 smoeq 6155 smores 6157 smores2 6159 smodm2 6160 smoiso 6167 tfrlem8 6183 tfri1dALT 6216 |
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