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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 4461 | . 2 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) | |
2 | disj3 3415 | . . 3 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐴})) | |
3 | df-suc 4293 | . . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
4 | 3 | difeq1i 3190 | . . . . 5 ⊢ (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴}) |
5 | difun2 3442 | . . . . 5 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = (𝐴 ∖ {𝐴}) | |
6 | 4, 5 | eqtri 2160 | . . . 4 ⊢ (suc 𝐴 ∖ {𝐴}) = (𝐴 ∖ {𝐴}) |
7 | 6 | eqeq2i 2150 | . . 3 ⊢ (𝐴 = (suc 𝐴 ∖ {𝐴}) ↔ 𝐴 = (𝐴 ∖ {𝐴})) |
8 | 2, 7 | bitr4i 186 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (suc 𝐴 ∖ {𝐴})) |
9 | 1, 8 | sylib 121 | 1 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∖ cdif 3068 ∪ cun 3069 ∩ cin 3070 ∅c0 3363 {csn 3527 Ord word 4284 suc csuc 4287 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-sn 3533 df-suc 4293 |
This theorem is referenced by: phplem3 6748 phplem4 6749 phplem4dom 6756 phplem4on 6761 dif1en 6773 |
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