Theorem elsuci 4415

Description: Lemma for ncfinraise 4482. 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.

Step	Hyp	Ref	Expression
1		elsuci.1	. . . . 5 ⊢ X ∈ V
2	1	elcompl 3226	. . . 4 ⊢ (X ∈ ∼ A ↔ ¬ X ∈ A)
3		eqid 2353	. . . . 5 ⊢ (A ∪ {X}) = (A ∪ {X})
4		sneq 3745	. . . . . . . 8 ⊢ (x = X → {x} = {X})
5	4	uneq2d 3419	. . . . . . 7 ⊢ (x = X → (A ∪ {x}) = (A ∪ {X}))
6	5	eqeq2d 2364	. . . . . 6 ⊢ (x = X → ((A ∪ {X}) = (A ∪ {x}) ↔ (A ∪ {X}) = (A ∪ {X})))
7	6	rspcev 2956	. . . . 5 ⊢ ((X ∈ ∼ A ∧ (A ∪ {X}) = (A ∪ {X})) → ∃x ∈ ∼ A(A ∪ {X}) = (A ∪ {x}))
8	3, 7	mpan2 652	. . . 4 ⊢ (X ∈ ∼ A → ∃x ∈ ∼ A(A ∪ {X}) = (A ∪ {x}))
9	2, 8	sylbir 204	. . 3 ⊢ (¬ X ∈ A → ∃x ∈ ∼ A(A ∪ {X}) = (A ∪ {x}))
10		compleq 3244	. . . . 5 ⊢ (a = A → ∼ a = ∼ A)
11		uneq1 3412	. . . . . 6 ⊢ (a = A → (a ∪ {x}) = (A ∪ {x}))
12	11	eqeq2d 2364	. . . . 5 ⊢ (a = A → ((A ∪ {X}) = (a ∪ {x}) ↔ (A ∪ {X}) = (A ∪ {x})))
13	10, 12	rexeqbidv 2821	. . . 4 ⊢ (a = A → (∃x ∈ ∼ a(A ∪ {X}) = (a ∪ {x}) ↔ ∃x ∈ ∼ A(A ∪ {X}) = (A ∪ {x})))
14	13	rspcev 2956	. . 3 ⊢ ((A ∈ N ∧ ∃x ∈ ∼ A(A ∪ {X}) = (A ∪ {x})) → ∃a ∈ N ∃x ∈ ∼ a(A ∪ {X}) = (a ∪ {x}))
15	9, 14	sylan2 460	. 2 ⊢ ((A ∈ N ∧ ¬ X ∈ A) → ∃a ∈ N ∃x ∈ ∼ a(A ∪ {X}) = (a ∪ {x}))
16		elsuc 4414	. 2 ⊢ ((A ∪ {X}) ∈ (N +c 1c) ↔ ∃a ∈ N ∃x ∈ ∼ a(A ∪ {X}) = (a ∪ {x}))
17	15, 16	sylibr 203	1 ⊢ ((A ∈ N ∧ ¬ X ∈ A) → (A ∪ {X}) ∈ (N +c 1c))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860   ∼ ccompl 3206   ∪ cun 3208  {csn 3738  1_cc1c 4135   +_c cplc 4376

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 4079  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-1c 4137  df-addc 4379

This theorem is referenced by:  prepeano4  4452  ncfinraise  4482  ncfinlower  4484  tfinsuc  4499  nnadjoin  4521  sfindbl  4531  tfinnn  4535  nulnnn  4557