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Theorem cantnflem2 7639
Description: Lemma for cantnf 7642. (Contributed by Mario Carneiro, 28-May-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( 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.1  |-  ( ph  ->  C  e.  ( A  ^o  B ) )
cantnf.2  |-  ( ph  ->  C  C_  ran  ( A CNF 
B ) )
cantnf.3  |-  ( 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.2 . . 3  |-  ( ph  ->  A  e.  On )
2 cantnfs.3 . . . . . . . . . 10  |-  ( ph  ->  B  e.  On )
3 oecl 6774 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
41, 2, 3syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
5 cantnf.1 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( A  ^o  B ) )
6 onelon 4599 . . . . . . . . 9  |-  ( ( ( A  ^o  B
)  e.  On  /\  C  e.  ( A  ^o  B ) )  ->  C  e.  On )
74, 5, 6syl2anc 643 . . . . . . . 8  |-  ( ph  ->  C  e.  On )
8 cantnf.3 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  C )
9 ondif1 6738 . . . . . . . 8  |-  ( C  e.  ( On  \  1o )  <->  ( C  e.  On  /\  (/)  e.  C
) )
107, 8, 9sylanbrc 646 . . . . . . 7  |-  ( ph  ->  C  e.  ( On 
\  1o ) )
1110eldifbd 3326 . . . . . 6  |-  ( ph  ->  -.  C  e.  1o )
12 ssel 3335 . . . . . . 7  |-  ( ( A  ^o  B ) 
C_  1o  ->  ( C  e.  ( A  ^o  B )  ->  C  e.  1o ) )
135, 12syl5com 28 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  B )  C_  1o  ->  C  e.  1o ) )
1411, 13mtod 170 . . . . 5  |-  ( ph  ->  -.  ( A  ^o  B )  C_  1o )
15 oe0m 6755 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
^o  B )  =  ( 1o  \  B
) )
162, 15syl 16 . . . . . . . 8  |-  ( ph  ->  ( (/)  ^o  B )  =  ( 1o  \  B ) )
17 difss 3467 . . . . . . . 8  |-  ( 1o 
\  B )  C_  1o
1816, 17syl6eqss 3391 . . . . . . 7  |-  ( ph  ->  ( (/)  ^o  B ) 
C_  1o )
19 oveq1 6081 . . . . . . . 8  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
2019sseq1d 3368 . . . . . . 7  |-  ( A  =  (/)  ->  ( ( A  ^o  B ) 
C_  1o  <->  ( (/)  ^o  B
)  C_  1o )
)
2118, 20syl5ibrcom 214 . . . . . 6  |-  ( ph  ->  ( A  =  (/)  ->  ( A  ^o  B
)  C_  1o )
)
22 oe1m 6781 . . . . . . . 8  |-  ( B  e.  On  ->  ( 1o  ^o  B )  =  1o )
23 eqimss 3393 . . . . . . . 8  |-  ( ( 1o  ^o  B )  =  1o  ->  ( 1o  ^o  B )  C_  1o )
242, 22, 233syl 19 . . . . . . 7  |-  ( ph  ->  ( 1o  ^o  B
)  C_  1o )
25 oveq1 6081 . . . . . . . 8  |-  ( A  =  1o  ->  ( A  ^o  B )  =  ( 1o  ^o  B
) )
2625sseq1d 3368 . . . . . . 7  |-  ( A  =  1o  ->  (
( A  ^o  B
)  C_  1o  <->  ( 1o  ^o  B )  C_  1o ) )
2724, 26syl5ibrcom 214 . . . . . 6  |-  ( ph  ->  ( A  =  1o 
->  ( A  ^o  B
)  C_  1o )
)
2821, 27jaod 370 . . . . 5  |-  ( ph  ->  ( ( A  =  (/)  \/  A  =  1o )  ->  ( A  ^o  B )  C_  1o ) )
2914, 28mtod 170 . . . 4  |-  ( ph  ->  -.  ( A  =  (/)  \/  A  =  1o ) )
30 elpri 3827 . . . . 5  |-  ( A  e.  { (/) ,  1o }  ->  ( A  =  (/)  \/  A  =  1o ) )
31 df2o3 6730 . . . . 5  |-  2o  =  { (/) ,  1o }
3230, 31eleq2s 2528 . . . 4  |-  ( A  e.  2o  ->  ( A  =  (/)  \/  A  =  1o ) )
3329, 32nsyl 115 . . 3  |-  ( ph  ->  -.  A  e.  2o )
341, 33eldifd 3324 . 2  |-  ( ph  ->  A  e.  ( On 
\  2o ) )
3534, 10jca 519 1  |-  ( ph  ->  ( A  e.  ( On  \  2o )  /\  C  e.  ( On  \  1o ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2698   E.wrex 2699    \ cdif 3310    C_ wss 3313   (/)c0 3621   {cpr 3808   {copab 4258   Oncon0 4574   dom cdm 4871   ran crn 4872   ` cfv 5447  (class class class)co 6074   1oc1o 6710   2oc2o 6711    ^o coe 6716   CNF ccnf 7609
This theorem is referenced by:  cantnflem3  7640  cantnflem4  7641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-recs 6626  df-rdg 6661  df-1o 6717  df-2o 6718  df-oadd 6721  df-omul 6722  df-oexp 6723
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