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Theorem cantnflem2 8002
Description: Lemma for cantnf 8005. (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 7080 . . . . . . . . . 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 4845 . . . . . . . . 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 7044 . . . . . . . 8  |-  ( C  e.  ( On  \  1o )  <->  ( C  e.  On  /\  (/)  e.  C
) )
107, 8, 9sylanbrc 664 . . . . . . 7  |-  ( ph  ->  C  e.  ( On 
\  1o ) )
1110eldifbd 3442 . . . . . 6  |-  ( ph  ->  -.  C  e.  1o )
12 ssel 3451 . . . . . . 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 7061 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
^o  B )  =  ( 1o  \  B
) )
162, 15syl 16 . . . . . . . 8  |-  ( ph  ->  ( (/)  ^o  B )  =  ( 1o  \  B ) )
17 difss 3584 . . . . . . . 8  |-  ( 1o 
\  B )  C_  1o
1816, 17syl6eqss 3507 . . . . . . 7  |-  ( ph  ->  ( (/)  ^o  B ) 
C_  1o )
19 oveq1 6200 . . . . . . . 8  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
2019sseq1d 3484 . . . . . . 7  |-  ( A  =  (/)  ->  ( ( A  ^o  B ) 
C_  1o  <->  ( (/)  ^o  B
)  C_  1o )
)
2118, 20syl5ibrcom 222 . . . . . 6  |-  ( ph  ->  ( A  =  (/)  ->  ( A  ^o  B
)  C_  1o )
)
22 oe1m 7087 . . . . . . . 8  |-  ( B  e.  On  ->  ( 1o  ^o  B )  =  1o )
23 eqimss 3509 . . . . . . . 8  |-  ( ( 1o  ^o  B )  =  1o  ->  ( 1o  ^o  B )  C_  1o )
242, 22, 233syl 20 . . . . . . 7  |-  ( ph  ->  ( 1o  ^o  B
)  C_  1o )
25 oveq1 6200 . . . . . . . 8  |-  ( A  =  1o  ->  ( A  ^o  B )  =  ( 1o  ^o  B
) )
2625sseq1d 3484 . . . . . . 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 3998 . . . . 5  |-  ( A  e.  { (/) ,  1o }  ->  ( A  =  (/)  \/  A  =  1o ) )
31 df2o3 7036 . . . . 5  |-  2o  =  { (/) ,  1o }
3230, 31eleq2s 2559 . . . 4  |-  ( A  e.  2o  ->  ( A  =  (/)  \/  A  =  1o ) )
3329, 32nsyl 121 . . 3  |-  ( ph  ->  -.  A  e.  2o )
341, 33eldifd 3440 . 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 1370    e. wcel 1758   A.wral 2795   E.wrex 2796    \ cdif 3426    C_ wss 3429   (/)c0 3738   {cpr 3980   {copab 4450   Oncon0 4820   dom cdm 4941   ran crn 4942   ` cfv 5519  (class class class)co 6193   1oc1o 7016   2oc2o 7017    ^o coe 7022   CNF ccnf 7971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-omul 7028  df-oexp 7029
This theorem is referenced by:  cantnflem3  8003  cantnflem4  8004  cantnflem3OLD  8025  cantnflem4OLD  8026
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