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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 X V
Assertion
Ref Expression
elsuci ((A N ¬ X A) → (A ∪ {X}) (N +c 1c))

Proof of Theorem elsuci
Dummy variables a x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elsuci.1 . . . . 5 X V
21elcompl 3225 . . . 4 (X A ↔ ¬ X A)
3 eqid 2353 . . . . 5 (A ∪ {X}) = (A ∪ {X})
4 sneq 3744 . . . . . . . 8 (x = X → {x} = {X})
54uneq2d 3418 . . . . . . 7 (x = X → (A ∪ {x}) = (A ∪ {X}))
65eqeq2d 2364 . . . . . 6 (x = X → ((A ∪ {X}) = (A ∪ {x}) ↔ (A ∪ {X}) = (A ∪ {X})))
76rspcev 2955 . . . . 5 ((X A (A ∪ {X}) = (A ∪ {X})) → x A(A ∪ {X}) = (A ∪ {x}))
83, 7mpan2 652 . . . 4 (X Ax A(A ∪ {X}) = (A ∪ {x}))
92, 8sylbir 204 . . 3 X Ax A(A ∪ {X}) = (A ∪ {x}))
10 compleq 3243 . . . . 5 (a = A → ∼ a = ∼ A)
11 uneq1 3411 . . . . . 6 (a = A → (a ∪ {x}) = (A ∪ {x}))
1211eqeq2d 2364 . . . . 5 (a = A → ((A ∪ {X}) = (a ∪ {x}) ↔ (A ∪ {X}) = (A ∪ {x})))
1310, 12rexeqbidv 2820 . . . 4 (a = A → (x a(A ∪ {X}) = (a ∪ {x}) ↔ x A(A ∪ {X}) = (A ∪ {x})))
1413rspcev 2955 . . 3 ((A N x A(A ∪ {X}) = (A ∪ {x})) → a N x a(A ∪ {X}) = (a ∪ {x}))
159, 14sylan2 460 . 2 ((A N ¬ X A) → a N x a(A ∪ {X}) = (a ∪ {x}))
16 elsuc 4413 . 2 ((A ∪ {X}) (N +c 1c) ↔ a N x a(A ∪ {X}) = (a ∪ {x}))
1715, 16sylibr 203 1 ((A N ¬ X A) → (A ∪ {X}) (N +c 1c))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859   ∼ ccompl 3205   ∪ cun 3207  {csn 3737  1cc1c 4134   +c cplc 4375 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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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