Theorem ennnfonelemk 12270

Description: Lemma for ennnfone 12295. (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 3192	. . . 4 |- ( N = K -> K C_ N )
3	2	adantl 275	. . 3 |- ( ( ph /\ N = K ) -> K C_ N )
4		eqid 2164	. . . . 5 |- ( F ` K ) = ( F ` K )
5		fveq2 5480	. . . . . . . . 9 |- ( j = K -> ( F ` j ) = ( F ` K ) )
6	5	neeq2d 2353	. . . . . . . 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 6451	. . . . . . . . . . 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 4566	. . . . . . . . . . 11 |- ( N e. om -> suc N e. om )
18		nnord 4583	. . . . . . . . . . 11 |- ( suc N e. om -> Ord suc N )
19	13, 17, 18	3syl 17	. . . . . . . . . 10 |- ( ( ph /\ K C_ N ) -> Ord suc N )
20		ordelsuc 4476	. . . . . . . . . 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 2830	. . . . . . 7 |- ( ( ph /\ K C_ N ) -> ( F ` K ) =/= ( F ` K ) )
24	23	neneqd 2355	. . . . . 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 634	. . . 4 |- ( ph -> -. K C_ N )
27	26	adantr 274	. . 3 |- ( ( ph /\ N = K ) -> -. K C_ N )
28	3, 27	pm2.21dd 610	. 2 |- ( ( ph /\ N = K ) -> N e. K )
29	12	adantr 274	. . . . 5 |- ( ( ph /\ K e. N ) -> N e. om )
30		nnon 4581	. . . . 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 4359	. . . 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 610	. 2 |- ( ( ph /\ K e. N ) -> N e. K )
37		nntri3or 6452	. . 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 1296	1 |- ( ph -> N e. K )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 966   = wceq 1342   e. wcel 2135   =/= wne 2334   A.wral 2442   C_ wss 3111   Ord word 4334   Oncon0 4335   suc csuc 4337   omcom 4561   -onto->wfo 5180   ` cfv 5182

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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-iinf 4559

This theorem depends on definitions:  df-bi 116  df-3or 968  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-v 2723  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-br 3977  df-tr 4075  df-iord 4338  df-on 4340  df-suc 4343  df-iom 4562  df-iota 5147  df-fv 5190

This theorem is referenced by:  ennnfonelemex  12284