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Theorem ennnfonelemk 11947
Description: Lemma for ennnfone 11972. (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 109 . 2  |-  ( (
ph  /\  N  e.  K )  ->  N  e.  K )
2 eqimss2 3156 . . . 4  |-  ( N  =  K  ->  K  C_  N )
32adantl 275 . . 3  |-  ( (
ph  /\  N  =  K )  ->  K  C_  N )
4 eqid 2140 . . . . 5  |-  ( F `
 K )  =  ( F `  K
)
5 fveq2 5428 . . . . . . . . 9  |-  ( j  =  K  ->  ( F `  j )  =  ( F `  K ) )
65neeq2d 2328 . . . . . . . 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 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 )
1110adantr 274 . . . . . . . . . . 11  |-  ( (
ph  /\  K  C_  N
)  ->  K  e.  om )
12 ennnfonelemk.n . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  om )
1312adantr 274 . . . . . . . . . . 11  |-  ( (
ph  /\  K  C_  N
)  ->  N  e.  om )
14 nnsucsssuc 6395 . . . . . . . . . . 11  |-  ( ( K  e.  om  /\  N  e.  om )  ->  ( K  C_  N  <->  suc 
K  C_  suc  N ) )
1511, 13, 14syl2anc 409 . . . . . . . . . 10  |-  ( (
ph  /\  K  C_  N
)  ->  ( K  C_  N  <->  suc  K  C_  suc  N ) )
169, 15mpbid 146 . . . . . . . . 9  |-  ( (
ph  /\  K  C_  N
)  ->  suc  K  C_  suc  N )
17 peano2 4516 . . . . . . . . . . 11  |-  ( N  e.  om  ->  suc  N  e.  om )
18 nnord 4532 . . . . . . . . . . 11  |-  ( suc 
N  e.  om  ->  Ord 
suc  N )
1913, 17, 183syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  K  C_  N
)  ->  Ord  suc  N
)
20 ordelsuc 4428 . . . . . . . . . 10  |-  ( ( K  e.  om  /\  Ord  suc  N )  -> 
( K  e.  suc  N  <->  suc  K  C_  suc  N ) )
2111, 19, 20syl2anc 409 . . . . . . . . 9  |-  ( (
ph  /\  K  C_  N
)  ->  ( K  e.  suc  N  <->  suc  K  C_  suc  N ) )
2216, 21mpbird 166 . . . . . . . 8  |-  ( (
ph  /\  K  C_  N
)  ->  K  e.  suc  N )
236, 8, 22rspcdva 2797 . . . . . . 7  |-  ( (
ph  /\  K  C_  N
)  ->  ( F `  K )  =/=  ( F `  K )
)
2423neneqd 2330 . . . . . 6  |-  ( (
ph  /\  K  C_  N
)  ->  -.  ( F `  K )  =  ( F `  K ) )
2524ex 114 . . . . 5  |-  ( ph  ->  ( K  C_  N  ->  -.  ( F `  K )  =  ( F `  K ) ) )
264, 25mt2i 634 . . . 4  |-  ( ph  ->  -.  K  C_  N
)
2726adantr 274 . . 3  |-  ( (
ph  /\  N  =  K )  ->  -.  K  C_  N )
283, 27pm2.21dd 610 . 2  |-  ( (
ph  /\  N  =  K )  ->  N  e.  K )
2912adantr 274 . . . . 5  |-  ( (
ph  /\  K  e.  N )  ->  N  e.  om )
30 nnon 4530 . . . . 5  |-  ( N  e.  om  ->  N  e.  On )
3129, 30syl 14 . . . 4  |-  ( (
ph  /\  K  e.  N )  ->  N  e.  On )
32 simpr 109 . . . 4  |-  ( (
ph  /\  K  e.  N )  ->  K  e.  N )
33 onelss 4316 . . . 4  |-  ( N  e.  On  ->  ( K  e.  N  ->  K 
C_  N ) )
3431, 32, 33sylc 62 . . 3  |-  ( (
ph  /\  K  e.  N )  ->  K  C_  N )
3526adantr 274 . . 3  |-  ( (
ph  /\  K  e.  N )  ->  -.  K  C_  N )
3634, 35pm2.21dd 610 . 2  |-  ( (
ph  /\  K  e.  N )  ->  N  e.  K )
37 nntri3or 6396 . . 3  |-  ( ( N  e.  om  /\  K  e.  om )  ->  ( N  e.  K  \/  N  =  K  \/  K  e.  N
) )
3812, 10, 37syl2anc 409 . 2  |-  ( ph  ->  ( N  e.  K  \/  N  =  K  \/  K  e.  N
) )
391, 28, 36, 38mpjao3dan 1286 1  |-  ( ph  ->  N  e.  K )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ w3o 962    = wceq 1332    e. wcel 1481    =/= wne 2309   A.wral 2417    C_ wss 3075   Ord word 4291   Oncon0 4292   suc csuc 4294   omcom 4511   -onto->wfo 5128   ` cfv 5130
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-br 3937  df-tr 4034  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-iota 5095  df-fv 5138
This theorem is referenced by:  ennnfonelemex  11961
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