Theorem enp1i 7670

Description: Proof induction for en2i 7472 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 4872	. . . . 5 |- suc M =/= (/)
2		breq1 4370	. . . . . . 7 |- ( A = (/) -> ( A ~~ N <-> (/) ~~ N ) )
3		enp1i.2	. . . . . . . 8 |- N = suc M
4		ensym 7483	. . . . . . . . 9 |- ( (/) ~~ N -> N ~~ (/) )
5		en0 7497	. . . . . . . . 9 |- ( N ~~ (/) <-> N = (/) )
6	4, 5	sylib 196	. . . . . . . 8 |- ( (/) ~~ N -> N = (/) )
7	3, 6	syl5eqr 2437	. . . . . . 7 |- ( (/) ~~ N -> suc M = (/) )
8	2, 7	syl6bi 228	. . . . . 6 |- ( A = (/) -> ( A ~~ N -> suc M = (/) ) )
9	8	necon3ad 2592	. . . . 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 3722	. . 3 |- ( -. A = (/) <-> E. x x e. A )
13	11, 12	sylib 196	. 2 |- ( A ~~ N -> E. x x e. A )
14	3	breq2i 4375	. . . . 5 |- ( A ~~ N <-> A ~~ suc M )
15		enp1i.1	. . . . . . . 8 |- M e. om
16		dif1en 7668	. . . . . . . 8 |- ( ( M e. om /\ A ~~ suc M /\ x e. A ) -> ( A \ { x } ) ~~ M )
17	15, 16	mp3an1 1309	. . . . . . 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 432	. . . . 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 1718	. 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 367   = wceq 1399   E.wex 1620   e. wcel 1826   =/= wne 2577   \ cdif 3386   (/)c0 3711   {csn 3944   class class class wbr 4367   suc csuc 4794   omcom 6599   ~~ cen 7432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-om 6600  df-1o 7048  df-er 7229  df-en 7436  df-fin 7439

This theorem is referenced by:  en2  7671  en3  7672  en4  7673