Theorem enp1i 7539

Description: Proof induction for en2i 7339 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.)

Hypotheses

Ref	Expression
enp1i.1	|- M e. om
enp1i.2	|- N = suc M
enp1i.3	|- ( ( A \ { x } ) ~~ M -> ph )
enp1i.4	|- ( x e. A -> ( ph -> ps ) )

Assertion

Ref	Expression
enp1i	|- ( A ~~ N -> E. x ps )

Distinct variable groups:   x, A   x, N

Allowed substitution hints:   ph( x)   ps( x)   M( x)

Proof of Theorem enp1i

Step	Hyp	Ref	Expression
1		nsuceq0 4794	. . . . 5 |- suc M =/= (/)
2		breq1 4290	. . . . . . 7 |- ( A = (/) -> ( A ~~ N <-> (/) ~~ N ) )
3		enp1i.2	. . . . . . . 8 |- N = suc M
4		ensym 7350	. . . . . . . . 9 |- ( (/) ~~ N -> N ~~ (/) )
5		en0 7364	. . . . . . . . 9 |- ( N ~~ (/) <-> N = (/) )
6	4, 5	sylib 196	. . . . . . . 8 |- ( (/) ~~ N -> N = (/) )
7	3, 6	syl5eqr 2484	. . . . . . 7 |- ( (/) ~~ N -> suc M = (/) )
8	2, 7	syl6bi 228	. . . . . 6 |- ( A = (/) -> ( A ~~ N -> suc M = (/) ) )
9	8	necon3ad 2639	. . . . 5 |- ( A = (/) -> ( suc M =/= (/) -> -. A ~~ N ) )
10	1, 9	mpi 17	. . . 4 |- ( A = (/) -> -. A ~~ N )
11	10	con2i 120	. . 3 |- ( A ~~ N -> -. A = (/) )
12		neq0 3642	. . 3 |- ( -. A = (/) <-> E. x x e. A )
13	11, 12	sylib 196	. 2 |- ( A ~~ N -> E. x x e. A )
14	3	breq2i 4295	. . . . 5 |- ( A ~~ N <-> A ~~ suc M )
15		enp1i.1	. . . . . . . 8 |- M e. om
16		dif1en 7537	. . . . . . . 8 |- ( ( M e. om /\ A ~~ suc M /\ x e. A ) -> ( A \ { x } ) ~~ M )
17	15, 16	mp3an1 1301	. . . . . . 7 |- ( ( A ~~ suc M /\ x e. A ) -> ( A \ { x } ) ~~ M )
18		enp1i.3	. . . . . . 7 |- ( ( A \ { x } ) ~~ M -> ph )
19	17, 18	syl 16	. . . . . 6 |- ( ( A ~~ suc M /\ x e. A ) -> ph )
20	19	ex 434	. . . . 5 |- ( A ~~ suc M -> ( x e. A -> ph ) )
21	14, 20	sylbi 195	. . . 4 |- ( A ~~ N -> ( x e. A -> ph ) )
22		enp1i.4	. . . 4 |- ( x e. A -> ( ph -> ps ) )
23	21, 22	sylcom 29	. . 3 |- ( A ~~ N -> ( x e. A -> ps ) )
24	23	eximdv 1676	. 2 |- ( A ~~ N -> ( E. x x e. A -> E. x ps ) )
25	13, 24	mpd 15	1 |- ( A ~~ N -> E. x ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1369   E.wex 1586   e. wcel 1756   =/= wne 2601   \ cdif 3320   (/)c0 3632   {csn 3872   class class class wbr 4287   suc csuc 4716   omcom 6471   ~~ cen 7299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-om 6472  df-1o 6912  df-er 7093  df-en 7303  df-fin 7306

This theorem is referenced by:  en2  7540  en3  7541  en4  7542