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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 3603 | . . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
3 | 1, 2 | uneq12d 3290 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ {𝐴}) = (𝐵 ∪ {𝐵})) |
4 | df-suc 4370 | . 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
5 | df-suc 4370 | . 2 ⊢ suc 𝐵 = (𝐵 ∪ {𝐵}) | |
6 | 3, 4, 5 | 3eqtr4g 2235 | 1 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∪ cun 3127 {csn 3592 suc csuc 4364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-sn 3598 df-suc 4370 |
This theorem is referenced by: eqelsuc 4418 2ordpr 4522 onsucsssucexmid 4525 onsucelsucexmid 4528 ordsucunielexmid 4529 suc11g 4555 onsucuni2 4562 0elsucexmid 4563 ordpwsucexmid 4568 peano2 4593 findes 4601 nn0suc 4602 0elnn 4617 omsinds 4620 tfr1onlemsucaccv 6339 tfrcllemsucaccv 6352 tfrcl 6362 frecabcl 6397 frecsuc 6405 sucinc 6443 sucinc2 6444 oacl 6458 oav2 6461 oasuc 6462 oa1suc 6465 nna0r 6476 nnacom 6482 nnaass 6483 nnmsucr 6486 nnsucelsuc 6489 nnsucsssuc 6490 nnaword 6509 nnaordex 6526 phplem3g 6853 nneneq 6854 php5 6855 php5dom 6860 omp1eomlem 7090 omp1eom 7091 nnnninfeq 7123 nnnninfeq2 7124 nninfwlpoimlemg 7170 nninfwlpoimlemginf 7171 nninfwlpoim 7173 indpi 7338 ennnfoneleminc 12404 ennnfonelemex 12407 bj-indsuc 14540 bj-bdfindes 14561 bj-nn0suc0 14562 bj-peano4 14567 bj-inf2vnlem1 14582 bj-nn0sucALT 14590 bj-findes 14593 nnsf 14614 nninfsellemdc 14619 nninfself 14622 nninfsellemeqinf 14625 nninfomni 14628 |
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