Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4270 | . 2 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) | |
2 | disj3 3272 | . . 3 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐴})) | |
3 | df-suc 4108 | . . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
4 | 3 | difeq1i 3058 | . . . . 5 ⊢ (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴}) |
5 | difun2 3302 | . . . . 5 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = (𝐴 ∖ {𝐴}) | |
6 | 4, 5 | eqtri 2060 | . . . 4 ⊢ (suc 𝐴 ∖ {𝐴}) = (𝐴 ∖ {𝐴}) |
7 | 6 | eqeq2i 2050 | . . 3 ⊢ (𝐴 = (suc 𝐴 ∖ {𝐴}) ↔ 𝐴 = (𝐴 ∖ {𝐴})) |
8 | 2, 7 | bitr4i 176 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (suc 𝐴 ∖ {𝐴})) |
9 | 1, 8 | sylib 127 | 1 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∖ cdif 2914 ∪ cun 2915 ∩ cin 2916 ∅c0 3224 {csn 3375 Ord word 4099 suc csuc 4102 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-sn 3381 df-suc 4108 |
This theorem is referenced by: phplem3 6317 phplem4 6318 phplem4dom 6324 phplem4on 6329 dif1en 6337 |
Copyright terms: Public domain | W3C validator |