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Theorem elsuci 4414
 Description: Lemma for ncfinraise 4481. Take a natural and a disjoint union and compute membership in the successor natural. (Contributed by SF, 22-Jan-2015.)
Hypothesis
Ref Expression
elsuci.1
Assertion
Ref Expression
elsuci 1c

Proof of Theorem elsuci
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elsuci.1 . . . . 5
21elcompl 3225 . . . 4
3 eqid 2353 . . . . 5
4 sneq 3744 . . . . . . . 8
54uneq2d 3418 . . . . . . 7
65eqeq2d 2364 . . . . . 6
76rspcev 2955 . . . . 5
83, 7mpan2 652 . . . 4
92, 8sylbir 204 . . 3
10 compleq 3243 . . . . 5
11 uneq1 3411 . . . . . 6
1211eqeq2d 2364 . . . . 5
1310, 12rexeqbidv 2820 . . . 4
1413rspcev 2955 . . 3
159, 14sylan2 460 . 2
16 elsuc 4413 . 2 1c
1715, 16sylibr 203 1 1c
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 358   wceq 1642   wcel 1710  wrex 2615  cvv 2859   ∼ ccompl 3205   cun 3207  csn 3737  1cc1c 4134   cplc 4375 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-1c 4136  df-addc 4378 This theorem is referenced by:  prepeano4  4451  ncfinraise  4481  ncfinlower  4483  tfinsuc  4498  nnadjoin  4520  sfindbl  4530  tfinnn  4534  nulnnn  4556
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