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Theorem ennnfonelemk 12371
Description: Lemma for ennnfone 12396. (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
StepHypRef Expression
1 simpr 110 . 2  |-  ( (
ph  /\  N  e.  K )  ->  N  e.  K )
2 eqimss2 3210 . . . 4  |-  ( N  =  K  ->  K  C_  N )
32adantl 277 . . 3  |-  ( (
ph  /\  N  =  K )  ->  K  C_  N )
4 eqid 2177 . . . . 5  |-  ( F `
 K )  =  ( F `  K
)
5 fveq2 5510 . . . . . . . . 9  |-  ( j  =  K  ->  ( F `  j )  =  ( F `  K ) )
65neeq2d 2366 . . . . . . . 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 ) )
87adantr 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 )
1110adantr 276 . . . . . . . . . . 11  |-  ( (
ph  /\  K  C_  N
)  ->  K  e.  om )
12 ennnfonelemk.n . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  om )
1312adantr 276 . . . . . . . . . . 11  |-  ( (
ph  /\  K  C_  N
)  ->  N  e.  om )
14 nnsucsssuc 6486 . . . . . . . . . . 11  |-  ( ( K  e.  om  /\  N  e.  om )  ->  ( K  C_  N  <->  suc 
K  C_  suc  N ) )
1511, 13, 14syl2anc 411 . . . . . . . . . 10  |-  ( (
ph  /\  K  C_  N
)  ->  ( K  C_  N  <->  suc  K  C_  suc  N ) )
169, 15mpbid 147 . . . . . . . . 9  |-  ( (
ph  /\  K  C_  N
)  ->  suc  K  C_  suc  N )
17 peano2 4590 . . . . . . . . . . 11  |-  ( N  e.  om  ->  suc  N  e.  om )
18 nnord 4607 . . . . . . . . . . 11  |-  ( suc 
N  e.  om  ->  Ord 
suc  N )
1913, 17, 183syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  K  C_  N
)  ->  Ord  suc  N
)
20 ordelsuc 4500 . . . . . . . . . 10  |-  ( ( K  e.  om  /\  Ord  suc  N )  -> 
( K  e.  suc  N  <->  suc  K  C_  suc  N ) )
2111, 19, 20syl2anc 411 . . . . . . . . 9  |-  ( (
ph  /\  K  C_  N
)  ->  ( K  e.  suc  N  <->  suc  K  C_  suc  N ) )
2216, 21mpbird 167 . . . . . . . 8  |-  ( (
ph  /\  K  C_  N
)  ->  K  e.  suc  N )
236, 8, 22rspcdva 2846 . . . . . . 7  |-  ( (
ph  /\  K  C_  N
)  ->  ( F `  K )  =/=  ( F `  K )
)
2423neneqd 2368 . . . . . 6  |-  ( (
ph  /\  K  C_  N
)  ->  -.  ( F `  K )  =  ( F `  K ) )
2524ex 115 . . . . 5  |-  ( ph  ->  ( K  C_  N  ->  -.  ( F `  K )  =  ( F `  K ) ) )
264, 25mt2i 644 . . . 4  |-  ( ph  ->  -.  K  C_  N
)
2726adantr 276 . . 3  |-  ( (
ph  /\  N  =  K )  ->  -.  K  C_  N )
283, 27pm2.21dd 620 . 2  |-  ( (
ph  /\  N  =  K )  ->  N  e.  K )
2912adantr 276 . . . . 5  |-  ( (
ph  /\  K  e.  N )  ->  N  e.  om )
30 nnon 4605 . . . . 5  |-  ( N  e.  om  ->  N  e.  On )
3129, 30syl 14 . . . 4  |-  ( (
ph  /\  K  e.  N )  ->  N  e.  On )
32 simpr 110 . . . 4  |-  ( (
ph  /\  K  e.  N )  ->  K  e.  N )
33 onelss 4383 . . . 4  |-  ( N  e.  On  ->  ( K  e.  N  ->  K 
C_  N ) )
3431, 32, 33sylc 62 . . 3  |-  ( (
ph  /\  K  e.  N )  ->  K  C_  N )
3526adantr 276 . . 3  |-  ( (
ph  /\  K  e.  N )  ->  -.  K  C_  N )
3634, 35pm2.21dd 620 . 2  |-  ( (
ph  /\  K  e.  N )  ->  N  e.  K )
37 nntri3or 6487 . . 3  |-  ( ( N  e.  om  /\  K  e.  om )  ->  ( N  e.  K  \/  N  =  K  \/  K  e.  N
) )
3812, 10, 37syl2anc 411 . 2  |-  ( ph  ->  ( N  e.  K  \/  N  =  K  \/  K  e.  N
) )
391, 28, 36, 38mpjao3dan 1307 1  |-  ( ph  ->  N  e.  K )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ w3o 977    = wceq 1353    e. wcel 2148    =/= wne 2347   A.wral 2455    C_ wss 3129   Ord word 4358   Oncon0 4359   suc csuc 4361   omcom 4585   -onto->wfo 5209   ` cfv 5211
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-iinf 4583
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-tr 4099  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-iota 5173  df-fv 5219
This theorem is referenced by:  ennnfonelemex  12385
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