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Mirrors > Home > ILE Home > Th. List > orddif | GIF version |
Description: Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
Ref | Expression |
---|---|
orddif | ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orddisj 4289 | . 2 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) | |
2 | disj3 3296 | . . 3 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐴})) | |
3 | df-suc 4126 | . . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
4 | 3 | difeq1i 3086 | . . . . 5 ⊢ (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴}) |
5 | difun2 3322 | . . . . 5 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = (𝐴 ∖ {𝐴}) | |
6 | 4, 5 | eqtri 2101 | . . . 4 ⊢ (suc 𝐴 ∖ {𝐴}) = (𝐴 ∖ {𝐴}) |
7 | 6 | eqeq2i 2091 | . . 3 ⊢ (𝐴 = (suc 𝐴 ∖ {𝐴}) ↔ 𝐴 = (𝐴 ∖ {𝐴})) |
8 | 2, 7 | bitr4i 185 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (suc 𝐴 ∖ {𝐴})) |
9 | 1, 8 | sylib 120 | 1 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∖ cdif 2970 ∪ cun 2971 ∩ cin 2972 ∅c0 3251 {csn 3398 Ord word 4117 suc csuc 4120 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-sn 3404 df-suc 4126 |
This theorem is referenced by: phplem3 6340 phplem4 6341 phplem4dom 6348 phplem4on 6353 dif1en 6364 |
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