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Mirrors > Home > ILE Home > Th. List > suceq | GIF version |
Description: Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
suceq | ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
2 | sneq 3508 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
3 | 1, 2 | uneq12d 3201 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ {𝐴}) = (𝐵 ∪ {𝐵})) |
4 | df-suc 4263 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | df-suc 4263 | . 2 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵}) | |
6 | 3, 4, 5 | 3eqtr4g 2175 | 1 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∪ cun 3039 {csn 3497 suc csuc 4257 |
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-v 2662 df-un 3045 df-sn 3503 df-suc 4263 |
This theorem is referenced by: eqelsuc 4311 2ordpr 4409 onsucsssucexmid 4412 onsucelsucexmid 4415 ordsucunielexmid 4416 suc11g 4442 onsucuni2 4449 0elsucexmid 4450 ordpwsucexmid 4455 peano2 4479 findes 4487 nn0suc 4488 0elnn 4502 omsinds 4505 tfr1onlemsucaccv 6206 tfrcllemsucaccv 6219 tfrcl 6229 frecabcl 6264 frecsuc 6272 sucinc 6309 sucinc2 6310 oacl 6324 oav2 6327 oasuc 6328 oa1suc 6331 nna0r 6342 nnacom 6348 nnaass 6349 nnmsucr 6352 nnsucelsuc 6355 nnsucsssuc 6356 nnaword 6375 nnaordex 6391 phplem3g 6718 nneneq 6719 php5 6720 php5dom 6725 omp1eomlem 6947 omp1eom 6948 indpi 7118 ennnfoneleminc 11851 ennnfonelemex 11854 bj-indsuc 13053 bj-bdfindes 13074 bj-nn0suc0 13075 bj-peano4 13080 bj-inf2vnlem1 13095 bj-nn0sucALT 13103 bj-findes 13106 nnsf 13126 nninfalllemn 13129 nninfsellemdc 13133 nninfself 13136 nninfsellemeqinf 13139 nninfomni 13142 |
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