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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 3571 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
3 | 1, 2 | uneq12d 3262 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ {𝐴}) = (𝐵 ∪ {𝐵})) |
4 | df-suc 4331 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | df-suc 4331 | . 2 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵}) | |
6 | 3, 4, 5 | 3eqtr4g 2215 | 1 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 ∪ cun 3100 {csn 3560 suc csuc 4325 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-un 3106 df-sn 3566 df-suc 4331 |
This theorem is referenced by: eqelsuc 4379 2ordpr 4483 onsucsssucexmid 4486 onsucelsucexmid 4489 ordsucunielexmid 4490 suc11g 4516 onsucuni2 4523 0elsucexmid 4524 ordpwsucexmid 4529 peano2 4554 findes 4562 nn0suc 4563 0elnn 4578 omsinds 4581 tfr1onlemsucaccv 6288 tfrcllemsucaccv 6301 tfrcl 6311 frecabcl 6346 frecsuc 6354 sucinc 6392 sucinc2 6393 oacl 6407 oav2 6410 oasuc 6411 oa1suc 6414 nna0r 6425 nnacom 6431 nnaass 6432 nnmsucr 6435 nnsucelsuc 6438 nnsucsssuc 6439 nnaword 6458 nnaordex 6474 phplem3g 6801 nneneq 6802 php5 6803 php5dom 6808 omp1eomlem 7038 omp1eom 7039 nnnninfeq 7071 nnnninfeq2 7072 indpi 7262 ennnfoneleminc 12151 ennnfonelemex 12154 bj-indsuc 13514 bj-bdfindes 13535 bj-nn0suc0 13536 bj-peano4 13541 bj-inf2vnlem1 13556 bj-nn0sucALT 13564 bj-findes 13567 nnsf 13588 nninfsellemdc 13593 nninfself 13596 nninfsellemeqinf 13599 nninfomni 13602 |
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