Theorem enp1i 7824

Description: Proof induction for en2i 7625 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 5510	. . . . 5 |- suc M =/= (/)
2		breq1 4398	. . . . . . 7 |- ( A = (/) -> ( A ~~ N <-> (/) ~~ N ) )
3		enp1i.2	. . . . . . . 8 |- N = suc M
4		ensym 7636	. . . . . . . . 9 |- ( (/) ~~ N -> N ~~ (/) )
5		en0 7650	. . . . . . . . 9 |- ( N ~~ (/) <-> N = (/) )
6	4, 5	sylib 201	. . . . . . . 8 |- ( (/) ~~ N -> N = (/) )
7	3, 6	syl5eqr 2519	. . . . . . 7 |- ( (/) ~~ N -> suc M = (/) )
8	2, 7	syl6bi 236	. . . . . 6 |- ( A = (/) -> ( A ~~ N -> suc M = (/) ) )
9	8	necon3ad 2656	. . . . 5 |- ( A = (/) -> ( suc M =/= (/) -> -. A ~~ N ) )
10	1, 9	mpi 20	. . . 4 |- ( A = (/) -> -. A ~~ N )
11	10	con2i 124	. . 3 |- ( A ~~ N -> -. A = (/) )
12		neq0 3733	. . 3 |- ( -. A = (/) <-> E. x x e. A )
13	11, 12	sylib 201	. 2 |- ( A ~~ N -> E. x x e. A )
14	3	breq2i 4403	. . . . 5 |- ( A ~~ N <-> A ~~ suc M )
15		enp1i.1	. . . . . . . 8 |- M e. om
16		dif1en 7822	. . . . . . . 8 |- ( ( M e. om /\ A ~~ suc M /\ x e. A ) -> ( A \ { x } ) ~~ M )
17	15, 16	mp3an1 1377	. . . . . . 7 |- ( ( A ~~ suc M /\ x e. A ) -> ( A \ { x } ) ~~ M )
18		enp1i.3	. . . . . . 7 |- ( ( A \ { x } ) ~~ M -> ph )
19	17, 18	syl 17	. . . . . 6 |- ( ( A ~~ suc M /\ x e. A ) -> ph )
20	19	ex 441	. . . . 5 |- ( A ~~ suc M -> ( x e. A -> ph ) )
21	14, 20	sylbi 200	. . . 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 1772	. 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 376   = wceq 1452   E.wex 1671   e. wcel 1904   =/= wne 2641   \ cdif 3387   (/)c0 3722   {csn 3959   class class class wbr 4395   suc csuc 5432   omcom 6711   ~~ cen 7584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-1o 7200  df-er 7381  df-en 7588  df-fin 7591

This theorem is referenced by:  en2  7825  en3  7826  en4  7827