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 e. _V

Assertion

Ref	Expression
elsuci	|- ((A e. N /\ -. X e. A) -> (A u. {X}) e. (N +o 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 e. _V
2	1	elcompl 3225	. . . 4 |- (X e. ∼ A <-> -. X e. A)
3		eqid 2353	. . . . 5 |- (A u. {X}) = (A u. {X})
4		sneq 3744	. . . . . . . 8 |- (x = X -> {x} = {X})
5	4	uneq2d 3418	. . . . . . 7 |- (x = X -> (A u. {x}) = (A u. {X}))
6	5	eqeq2d 2364	. . . . . 6 |- (x = X -> ((A u. {X}) = (A u. {x}) <-> (A u. {X}) = (A u. {X})))
7	6	rspcev 2955	. . . . 5 |- ((X e. ∼ A /\ (A u. {X}) = (A u. {X})) -> E.x e. ∼ A(A u. {X}) = (A u. {x}))
8	3, 7	mpan2 652	. . . 4 |- (X e. ∼ A -> E.x e. ∼ A(A u. {X}) = (A u. {x}))
9	2, 8	sylbir 204	. . 3 |- (-. X e. A -> E.x e. ∼ A(A u. {X}) = (A u. {x}))
10		compleq 3243	. . . . 5 |- (a = A -> ∼ a = ∼ A)
11		uneq1 3411	. . . . . 6 |- (a = A -> (a u. {x}) = (A u. {x}))
12	11	eqeq2d 2364	. . . . 5 |- (a = A -> ((A u. {X}) = (a u. {x}) <-> (A u. {X}) = (A u. {x})))
13	10, 12	rexeqbidv 2820	. . . 4 |- (a = A -> (E.x e. ∼ a(A u. {X}) = (a u. {x}) <-> E.x e. ∼ A(A u. {X}) = (A u. {x})))
14	13	rspcev 2955	. . 3 |- ((A e. N /\ E.x e. ∼ A(A u. {X}) = (A u. {x})) -> E.a e. N E.x e. ∼ a(A u. {X}) = (a u. {x}))
15	9, 14	sylan2 460	. 2 |- ((A e. N /\ -. X e. A) -> E.a e. N E.x e. ∼ a(A u. {X}) = (a u. {x}))
16		elsuc 4413	. 2 |- ((A u. {X}) e. (N +o 1c) <-> E.a e. N E.x e. ∼ a(A u. {X}) = (a u. {x}))
17	15, 16	sylibr 203	1 |- ((A e. N /\ -. X e. A) -> (A u. {X}) e. (N +o 1c))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2615  _Vcvv 2859   ∼ ccompl 3205   u. cun 3207  {csn 3737  1_cc1c 4134   +o 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