Theorem ennnfonelemk 12557

Description: Lemma for ennnfone 12582. (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 110	. 2 |- ( ( ph /\ N e. K ) -> N e. K )
2		eqimss2 3234	. . . 4 |- ( N = K -> K C_ N )
3	2	adantl 277	. . 3 |- ( ( ph /\ N = K ) -> K C_ N )
4		eqid 2193	. . . . 5 |- ( F ` K ) = ( F ` K )
5		fveq2 5554	. . . . . . . . 9 |- ( j = K -> ( F ` j ) = ( F ` K ) )
6	5	neeq2d 2383	. . . . . . . 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 276	. . . . . . . 8 |- ( ( ph /\ K C_ N ) -> A. j e. suc N ( F ` K ) =/= ( F ` j ) )
9		simpr 110	. . . . . . . . . 10 |- ( ( ph /\ K C_ N ) -> K C_ N )
10		ennnfonelemk.k	. . . . . . . . . . . 12 |- ( ph -> K e. om )
11	10	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ K C_ N ) -> K e. om )
12		ennnfonelemk.n	. . . . . . . . . . . 12 |- ( ph -> N e. om )
13	12	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ K C_ N ) -> N e. om )
14		nnsucsssuc 6545	. . . . . . . . . . 11 |- ( ( K e. om /\ N e. om ) -> ( K C_ N <-> suc K C_ suc N ) )
15	11, 13, 14	syl2anc 411	. . . . . . . . . 10 |- ( ( ph /\ K C_ N ) -> ( K C_ N <-> suc K C_ suc N ) )
16	9, 15	mpbid 147	. . . . . . . . 9 |- ( ( ph /\ K C_ N ) -> suc K C_ suc N )
17		peano2 4627	. . . . . . . . . . 11 |- ( N e. om -> suc N e. om )
18		nnord 4644	. . . . . . . . . . 11 |- ( suc N e. om -> Ord suc N )
19	13, 17, 18	3syl 17	. . . . . . . . . 10 |- ( ( ph /\ K C_ N ) -> Ord suc N )
20		ordelsuc 4537	. . . . . . . . . 10 |- ( ( K e. om /\ Ord suc N ) -> ( K e. suc N <-> suc K C_ suc N ) )
21	11, 19, 20	syl2anc 411	. . . . . . . . 9 |- ( ( ph /\ K C_ N ) -> ( K e. suc N <-> suc K C_ suc N ) )
22	16, 21	mpbird 167	. . . . . . . 8 |- ( ( ph /\ K C_ N ) -> K e. suc N )
23	6, 8, 22	rspcdva 2869	. . . . . . 7 |- ( ( ph /\ K C_ N ) -> ( F ` K ) =/= ( F ` K ) )
24	23	neneqd 2385	. . . . . 6 |- ( ( ph /\ K C_ N ) -> -. ( F ` K ) = ( F ` K ) )
25	24	ex 115	. . . . 5 |- ( ph -> ( K C_ N -> -. ( F ` K ) = ( F ` K ) ) )
26	4, 25	mt2i 645	. . . 4 |- ( ph -> -. K C_ N )
27	26	adantr 276	. . 3 |- ( ( ph /\ N = K ) -> -. K C_ N )
28	3, 27	pm2.21dd 621	. 2 |- ( ( ph /\ N = K ) -> N e. K )
29	12	adantr 276	. . . . 5 |- ( ( ph /\ K e. N ) -> N e. om )
30		nnon 4642	. . . . 5 |- ( N e. om -> N e. On )
31	29, 30	syl 14	. . . 4 |- ( ( ph /\ K e. N ) -> N e. On )
32		simpr 110	. . . 4 |- ( ( ph /\ K e. N ) -> K e. N )
33		onelss 4418	. . . 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 276	. . 3 |- ( ( ph /\ K e. N ) -> -. K C_ N )
36	34, 35	pm2.21dd 621	. 2 |- ( ( ph /\ K e. N ) -> N e. K )
37		nntri3or 6546	. . 3 |- ( ( N e. om /\ K e. om ) -> ( N e. K \/ N = K \/ K e. N ) )
38	12, 10, 37	syl2anc 411	. 2 |- ( ph -> ( N e. K \/ N = K \/ K e. N ) )
39	1, 28, 36, 38	mpjao3dan 1318	1 |- ( ph -> N e. K )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 979   = wceq 1364   e. wcel 2164   =/= wne 2364   A.wral 2472   C_ wss 3153   Ord word 4393   Oncon0 4394   suc csuc 4396   omcom 4622   -onto->wfo 5252   ` cfv 5254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-tr 4128  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-iota 5215  df-fv 5262

This theorem is referenced by:  ennnfonelemex  12571