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Theorem ennnfonelemk 12414
Description: Lemma for ennnfone 12439. (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 3222 . . . 4  |-  ( N  =  K  ->  K  C_  N )
32adantl 277 . . 3  |-  ( (
ph  /\  N  =  K )  ->  K  C_  N )
4 eqid 2187 . . . . 5  |-  ( F `
 K )  =  ( F `  K
)
5 fveq2 5527 . . . . . . . . 9  |-  ( j  =  K  ->  ( F `  j )  =  ( F `  K ) )
65neeq2d 2376 . . . . . . . 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 6506 . . . . . . . . . . 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 4606 . . . . . . . . . . 11  |-  ( N  e.  om  ->  suc  N  e.  om )
18 nnord 4623 . . . . . . . . . . 11  |-  ( suc 
N  e.  om  ->  Ord 
suc  N )
1913, 17, 183syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  K  C_  N
)  ->  Ord  suc  N
)
20 ordelsuc 4516 . . . . . . . . . 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 2858 . . . . . . 7  |-  ( (
ph  /\  K  C_  N
)  ->  ( F `  K )  =/=  ( F `  K )
)
2423neneqd 2378 . . . . . 6  |-  ( (
ph  /\  K  C_  N
)  ->  -.  ( F `  K )  =  ( F `  K ) )
2524ex 115 . . . . 5  |-  ( ph  ->  ( K  C_  N  ->  -.  ( F `  K )  =  ( F `  K ) ) )
264, 25mt2i 645 . . . 4  |-  ( ph  ->  -.  K  C_  N
)
2726adantr 276 . . 3  |-  ( (
ph  /\  N  =  K )  ->  -.  K  C_  N )
283, 27pm2.21dd 621 . 2  |-  ( (
ph  /\  N  =  K )  ->  N  e.  K )
2912adantr 276 . . . . 5  |-  ( (
ph  /\  K  e.  N )  ->  N  e.  om )
30 nnon 4621 . . . . 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 4399 . . . 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 621 . 2  |-  ( (
ph  /\  K  e.  N )  ->  N  e.  K )
37 nntri3or 6507 . . 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 1317 1  |-  ( ph  ->  N  e.  K )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ w3o 978    = wceq 1363    e. wcel 2158    =/= wne 2357   A.wral 2465    C_ wss 3141   Ord word 4374   Oncon0 4375   suc csuc 4377   omcom 4601   -onto->wfo 5226   ` cfv 5228
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-tr 4114  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-iota 5190  df-fv 5236
This theorem is referenced by:  ennnfonelemex  12428
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