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Theorem cantnflem2 8126
Description: Lemma for cantnf 8129. (Contributed by Mario Carneiro, 28-May-2015.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
oemapval.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
cantnf.c  |-  ( ph  ->  C  e.  ( A  ^o  B ) )
cantnf.s  |-  ( ph  ->  C  C_  ran  ( A CNF 
B ) )
cantnf.e  |-  ( ph  -> 
(/)  e.  C )
Assertion
Ref Expression
cantnflem2  |-  ( ph  ->  ( A  e.  ( On  \  2o )  /\  C  e.  ( On  \  1o ) ) )
Distinct variable groups:    x, w, y, z, B    w, C, x, y, z    w, A, x, y, z    x, S, y, z    ph, x, y, z
Allowed substitution hints:    ph( w)    S( w)    T( x, y, z, w)

Proof of Theorem cantnflem2
StepHypRef Expression
1 cantnfs.a . . 3  |-  ( ph  ->  A  e.  On )
2 cantnfs.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  On )
3 oecl 7205 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
41, 2, 3syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
5 cantnf.c . . . . . . . . 9  |-  ( ph  ->  C  e.  ( A  ^o  B ) )
6 onelon 4912 . . . . . . . . 9  |-  ( ( ( A  ^o  B
)  e.  On  /\  C  e.  ( A  ^o  B ) )  ->  C  e.  On )
74, 5, 6syl2anc 661 . . . . . . . 8  |-  ( ph  ->  C  e.  On )
8 cantnf.e . . . . . . . 8  |-  ( ph  -> 
(/)  e.  C )
9 ondif1 7169 . . . . . . . 8  |-  ( C  e.  ( On  \  1o )  <->  ( C  e.  On  /\  (/)  e.  C
) )
107, 8, 9sylanbrc 664 . . . . . . 7  |-  ( ph  ->  C  e.  ( On 
\  1o ) )
1110eldifbd 3484 . . . . . 6  |-  ( ph  ->  -.  C  e.  1o )
12 ssel 3493 . . . . . . 7  |-  ( ( A  ^o  B ) 
C_  1o  ->  ( C  e.  ( A  ^o  B )  ->  C  e.  1o ) )
135, 12syl5com 30 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  B )  C_  1o  ->  C  e.  1o ) )
1411, 13mtod 177 . . . . 5  |-  ( ph  ->  -.  ( A  ^o  B )  C_  1o )
15 oe0m 7186 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
^o  B )  =  ( 1o  \  B
) )
162, 15syl 16 . . . . . . . 8  |-  ( ph  ->  ( (/)  ^o  B )  =  ( 1o  \  B ) )
17 difss 3627 . . . . . . . 8  |-  ( 1o 
\  B )  C_  1o
1816, 17syl6eqss 3549 . . . . . . 7  |-  ( ph  ->  ( (/)  ^o  B ) 
C_  1o )
19 oveq1 6303 . . . . . . . 8  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
2019sseq1d 3526 . . . . . . 7  |-  ( A  =  (/)  ->  ( ( A  ^o  B ) 
C_  1o  <->  ( (/)  ^o  B
)  C_  1o )
)
2118, 20syl5ibrcom 222 . . . . . 6  |-  ( ph  ->  ( A  =  (/)  ->  ( A  ^o  B
)  C_  1o )
)
22 oe1m 7212 . . . . . . . 8  |-  ( B  e.  On  ->  ( 1o  ^o  B )  =  1o )
23 eqimss 3551 . . . . . . . 8  |-  ( ( 1o  ^o  B )  =  1o  ->  ( 1o  ^o  B )  C_  1o )
242, 22, 233syl 20 . . . . . . 7  |-  ( ph  ->  ( 1o  ^o  B
)  C_  1o )
25 oveq1 6303 . . . . . . . 8  |-  ( A  =  1o  ->  ( A  ^o  B )  =  ( 1o  ^o  B
) )
2625sseq1d 3526 . . . . . . 7  |-  ( A  =  1o  ->  (
( A  ^o  B
)  C_  1o  <->  ( 1o  ^o  B )  C_  1o ) )
2724, 26syl5ibrcom 222 . . . . . 6  |-  ( ph  ->  ( A  =  1o 
->  ( A  ^o  B
)  C_  1o )
)
2821, 27jaod 380 . . . . 5  |-  ( ph  ->  ( ( A  =  (/)  \/  A  =  1o )  ->  ( A  ^o  B )  C_  1o ) )
2914, 28mtod 177 . . . 4  |-  ( ph  ->  -.  ( A  =  (/)  \/  A  =  1o ) )
30 elpri 4052 . . . . 5  |-  ( A  e.  { (/) ,  1o }  ->  ( A  =  (/)  \/  A  =  1o ) )
31 df2o3 7161 . . . . 5  |-  2o  =  { (/) ,  1o }
3230, 31eleq2s 2565 . . . 4  |-  ( A  e.  2o  ->  ( A  =  (/)  \/  A  =  1o ) )
3329, 32nsyl 121 . . 3  |-  ( ph  ->  -.  A  e.  2o )
341, 33eldifd 3482 . 2  |-  ( ph  ->  A  e.  ( On 
\  2o ) )
3534, 10jca 532 1  |-  ( ph  ->  ( A  e.  ( On  \  2o )  /\  C  e.  ( On  \  1o ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    \ cdif 3468    C_ wss 3471   (/)c0 3793   {cpr 4034   {copab 4514   Oncon0 4887   dom cdm 5008   ran crn 5009   ` cfv 5594  (class class class)co 6296   1oc1o 7141   2oc2o 7142    ^o coe 7147   CNF ccnf 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-oexp 7154
This theorem is referenced by:  cantnflem3  8127  cantnflem4  8128  cantnflem3OLD  8149  cantnflem4OLD  8150
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