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Theorem cantnflem2 8200
Description: Lemma for cantnf 8203. (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 7244 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
41, 2, 3syl2anc 667 . . . . . . . . 9  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
5 cantnf.c . . . . . . . . 9  |-  ( ph  ->  C  e.  ( A  ^o  B ) )
6 onelon 5451 . . . . . . . . 9  |-  ( ( ( A  ^o  B
)  e.  On  /\  C  e.  ( A  ^o  B ) )  ->  C  e.  On )
74, 5, 6syl2anc 667 . . . . . . . 8  |-  ( ph  ->  C  e.  On )
8 cantnf.e . . . . . . . 8  |-  ( ph  -> 
(/)  e.  C )
9 ondif1 7208 . . . . . . . 8  |-  ( C  e.  ( On  \  1o )  <->  ( C  e.  On  /\  (/)  e.  C
) )
107, 8, 9sylanbrc 671 . . . . . . 7  |-  ( ph  ->  C  e.  ( On 
\  1o ) )
1110eldifbd 3419 . . . . . 6  |-  ( ph  ->  -.  C  e.  1o )
12 ssel 3428 . . . . . . 7  |-  ( ( A  ^o  B ) 
C_  1o  ->  ( C  e.  ( A  ^o  B )  ->  C  e.  1o ) )
135, 12syl5com 31 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  B )  C_  1o  ->  C  e.  1o ) )
1411, 13mtod 181 . . . . 5  |-  ( ph  ->  -.  ( A  ^o  B )  C_  1o )
15 oe0m 7225 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
^o  B )  =  ( 1o  \  B
) )
162, 15syl 17 . . . . . . . 8  |-  ( ph  ->  ( (/)  ^o  B )  =  ( 1o  \  B ) )
17 difss 3562 . . . . . . . 8  |-  ( 1o 
\  B )  C_  1o
1816, 17syl6eqss 3484 . . . . . . 7  |-  ( ph  ->  ( (/)  ^o  B ) 
C_  1o )
19 oveq1 6302 . . . . . . . 8  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
2019sseq1d 3461 . . . . . . 7  |-  ( A  =  (/)  ->  ( ( A  ^o  B ) 
C_  1o  <->  ( (/)  ^o  B
)  C_  1o )
)
2118, 20syl5ibrcom 226 . . . . . 6  |-  ( ph  ->  ( A  =  (/)  ->  ( A  ^o  B
)  C_  1o )
)
22 oe1m 7251 . . . . . . . 8  |-  ( B  e.  On  ->  ( 1o  ^o  B )  =  1o )
23 eqimss 3486 . . . . . . . 8  |-  ( ( 1o  ^o  B )  =  1o  ->  ( 1o  ^o  B )  C_  1o )
242, 22, 233syl 18 . . . . . . 7  |-  ( ph  ->  ( 1o  ^o  B
)  C_  1o )
25 oveq1 6302 . . . . . . . 8  |-  ( A  =  1o  ->  ( A  ^o  B )  =  ( 1o  ^o  B
) )
2625sseq1d 3461 . . . . . . 7  |-  ( A  =  1o  ->  (
( A  ^o  B
)  C_  1o  <->  ( 1o  ^o  B )  C_  1o ) )
2724, 26syl5ibrcom 226 . . . . . 6  |-  ( ph  ->  ( A  =  1o 
->  ( A  ^o  B
)  C_  1o )
)
2821, 27jaod 382 . . . . 5  |-  ( ph  ->  ( ( A  =  (/)  \/  A  =  1o )  ->  ( A  ^o  B )  C_  1o ) )
2914, 28mtod 181 . . . 4  |-  ( ph  ->  -.  ( A  =  (/)  \/  A  =  1o ) )
30 elpri 3987 . . . . 5  |-  ( A  e.  { (/) ,  1o }  ->  ( A  =  (/)  \/  A  =  1o ) )
31 df2o3 7200 . . . . 5  |-  2o  =  { (/) ,  1o }
3230, 31eleq2s 2549 . . . 4  |-  ( A  e.  2o  ->  ( A  =  (/)  \/  A  =  1o ) )
3329, 32nsyl 125 . . 3  |-  ( ph  ->  -.  A  e.  2o )
341, 33eldifd 3417 . 2  |-  ( ph  ->  A  e.  ( On 
\  2o ) )
3534, 10jca 535 1  |-  ( ph  ->  ( A  e.  ( On  \  2o )  /\  C  e.  ( On  \  1o ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 370    /\ wa 371    = wceq 1446    e. wcel 1889   A.wral 2739   E.wrex 2740    \ cdif 3403    C_ wss 3406   (/)c0 3733   {cpr 3972   {copab 4463   dom cdm 4837   ran crn 4838   Oncon0 5426   ` cfv 5585  (class class class)co 6295   1oc1o 7180   2oc2o 7181    ^o coe 7186   CNF ccnf 8171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-omul 7192  df-oexp 7193
This theorem is referenced by:  cantnflem3  8201  cantnflem4  8202
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