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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 3594 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
3 | 1, 2 | uneq12d 3282 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ {𝐴}) = (𝐵 ∪ {𝐵})) |
4 | df-suc 4356 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | df-suc 4356 | . 2 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵}) | |
6 | 3, 4, 5 | 3eqtr4g 2228 | 1 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∪ cun 3119 {csn 3583 suc csuc 4350 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-sn 3589 df-suc 4356 |
This theorem is referenced by: eqelsuc 4404 2ordpr 4508 onsucsssucexmid 4511 onsucelsucexmid 4514 ordsucunielexmid 4515 suc11g 4541 onsucuni2 4548 0elsucexmid 4549 ordpwsucexmid 4554 peano2 4579 findes 4587 nn0suc 4588 0elnn 4603 omsinds 4606 tfr1onlemsucaccv 6320 tfrcllemsucaccv 6333 tfrcl 6343 frecabcl 6378 frecsuc 6386 sucinc 6424 sucinc2 6425 oacl 6439 oav2 6442 oasuc 6443 oa1suc 6446 nna0r 6457 nnacom 6463 nnaass 6464 nnmsucr 6467 nnsucelsuc 6470 nnsucsssuc 6471 nnaword 6490 nnaordex 6507 phplem3g 6834 nneneq 6835 php5 6836 php5dom 6841 omp1eomlem 7071 omp1eom 7072 nnnninfeq 7104 nnnninfeq2 7105 nninfwlpoimlemg 7151 nninfwlpoimlemginf 7152 nninfwlpoim 7154 indpi 7304 ennnfoneleminc 12366 ennnfonelemex 12369 bj-indsuc 13963 bj-bdfindes 13984 bj-nn0suc0 13985 bj-peano4 13990 bj-inf2vnlem1 14005 bj-nn0sucALT 14013 bj-findes 14016 nnsf 14038 nninfsellemdc 14043 nninfself 14046 nninfsellemeqinf 14049 nninfomni 14052 |
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