Theorem ennnfonelemk 12355

Description: Lemma for ennnfone 12380. (Contributed by Jim Kingdon, 15-Jul-2023.)

Hypotheses

Ref	Expression
ennnfonelemk.f	|- ( ph -> F : om -onto-> A )
ennnfonelemk.k	|- ( ph -> K e. om )
ennnfonelemk.n	|- ( ph -> N e. om )
ennnfonelemk.j	|- ( ph -> A. j e. suc N ( F ` K ) =/= ( F ` j ) )

Assertion

Ref	Expression
ennnfonelemk	|- ( ph -> N e. K )

Distinct variable groups:   j, F   j, K   j, N

Allowed substitution hints:   ph( j)   A( j)

Proof of Theorem ennnfonelemk

Step	Hyp	Ref	Expression
1		simpr 109	. 2 |- ( ( ph /\ N e. K ) -> N e. K )
2		eqimss2 3202	. . . 4 |- ( N = K -> K C_ N )
3	2	adantl 275	. . 3 |- ( ( ph /\ N = K ) -> K C_ N )
4		eqid 2170	. . . . 5 |- ( F ` K ) = ( F ` K )
5		fveq2 5496	. . . . . . . . 9 |- ( j = K -> ( F ` j ) = ( F ` K ) )
6	5	neeq2d 2359	. . . . . . . 8 |- ( j = K -> ( ( F ` K ) =/= ( F ` j ) <-> ( F ` K ) =/= ( F ` K ) ) )
7		ennnfonelemk.j	. . . . . . . . 9 |- ( ph -> A. j e. suc N ( F ` K ) =/= ( F ` j ) )
8	7	adantr 274	. . . . . . . 8 |- ( ( ph /\ K C_ N ) -> A. j e. suc N ( F ` K ) =/= ( F ` j ) )
9		simpr 109	. . . . . . . . . 10 |- ( ( ph /\ K C_ N ) -> K C_ N )
10		ennnfonelemk.k	. . . . . . . . . . . 12 |- ( ph -> K e. om )
11	10	adantr 274	. . . . . . . . . . 11 |- ( ( ph /\ K C_ N ) -> K e. om )
12		ennnfonelemk.n	. . . . . . . . . . . 12 |- ( ph -> N e. om )
13	12	adantr 274	. . . . . . . . . . 11 |- ( ( ph /\ K C_ N ) -> N e. om )
14		nnsucsssuc 6471	. . . . . . . . . . 11 |- ( ( K e. om /\ N e. om ) -> ( K C_ N <-> suc K C_ suc N ) )
15	11, 13, 14	syl2anc 409	. . . . . . . . . 10 |- ( ( ph /\ K C_ N ) -> ( K C_ N <-> suc K C_ suc N ) )
16	9, 15	mpbid 146	. . . . . . . . 9 |- ( ( ph /\ K C_ N ) -> suc K C_ suc N )
17		peano2 4579	. . . . . . . . . . 11 |- ( N e. om -> suc N e. om )
18		nnord 4596	. . . . . . . . . . 11 |- ( suc N e. om -> Ord suc N )
19	13, 17, 18	3syl 17	. . . . . . . . . 10 |- ( ( ph /\ K C_ N ) -> Ord suc N )
20		ordelsuc 4489	. . . . . . . . . 10 |- ( ( K e. om /\ Ord suc N ) -> ( K e. suc N <-> suc K C_ suc N ) )
21	11, 19, 20	syl2anc 409	. . . . . . . . 9 |- ( ( ph /\ K C_ N ) -> ( K e. suc N <-> suc K C_ suc N ) )
22	16, 21	mpbird 166	. . . . . . . 8 |- ( ( ph /\ K C_ N ) -> K e. suc N )
23	6, 8, 22	rspcdva 2839	. . . . . . 7 |- ( ( ph /\ K C_ N ) -> ( F ` K ) =/= ( F ` K ) )
24	23	neneqd 2361	. . . . . 6 |- ( ( ph /\ K C_ N ) -> -. ( F ` K ) = ( F ` K ) )
25	24	ex 114	. . . . 5 |- ( ph -> ( K C_ N -> -. ( F ` K ) = ( F ` K ) ) )
26	4, 25	mt2i 639	. . . 4 |- ( ph -> -. K C_ N )
27	26	adantr 274	. . 3 |- ( ( ph /\ N = K ) -> -. K C_ N )
28	3, 27	pm2.21dd 615	. 2 |- ( ( ph /\ N = K ) -> N e. K )
29	12	adantr 274	. . . . 5 |- ( ( ph /\ K e. N ) -> N e. om )
30		nnon 4594	. . . . 5 |- ( N e. om -> N e. On )
31	29, 30	syl 14	. . . 4 |- ( ( ph /\ K e. N ) -> N e. On )
32		simpr 109	. . . 4 |- ( ( ph /\ K e. N ) -> K e. N )
33		onelss 4372	. . . 4 |- ( N e. On -> ( K e. N -> K C_ N ) )
34	31, 32, 33	sylc 62	. . 3 |- ( ( ph /\ K e. N ) -> K C_ N )
35	26	adantr 274	. . 3 |- ( ( ph /\ K e. N ) -> -. K C_ N )
36	34, 35	pm2.21dd 615	. 2 |- ( ( ph /\ K e. N ) -> N e. K )
37		nntri3or 6472	. . 3 |- ( ( N e. om /\ K e. om ) -> ( N e. K \/ N = K \/ K e. N ) )
38	12, 10, 37	syl2anc 409	. 2 |- ( ph -> ( N e. K \/ N = K \/ K e. N ) )
39	1, 28, 36, 38	mpjao3dan 1302	1 |- ( ph -> N e. K )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 972   = wceq 1348   e. wcel 2141   =/= wne 2340   A.wral 2448   C_ wss 3121   Ord word 4347   Oncon0 4348   suc csuc 4350   omcom 4574   -onto->wfo 5196   ` cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-iota 5160  df-fv 5206

This theorem is referenced by:  ennnfonelemex  12369