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Theorem cantnflem2 7408
Description: Lemma for cantnf 7411. (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 6552 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
41, 2, 3syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
5 cantnf.1 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( A  ^o  B ) )
6 onelon 4433 . . . . . . . . 9  |-  ( ( ( A  ^o  B
)  e.  On  /\  C  e.  ( A  ^o  B ) )  ->  C  e.  On )
74, 5, 6syl2anc 642 . . . . . . . 8  |-  ( ph  ->  C  e.  On )
8 cantnf.3 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  C )
9 ondif1 6516 . . . . . . . 8  |-  ( C  e.  ( On  \  1o )  <->  ( C  e.  On  /\  (/)  e.  C
) )
107, 8, 9sylanbrc 645 . . . . . . 7  |-  ( ph  ->  C  e.  ( On 
\  1o ) )
11 eldifn 3312 . . . . . . 7  |-  ( C  e.  ( On  \  1o )  ->  -.  C  e.  1o )
1210, 11syl 15 . . . . . 6  |-  ( ph  ->  -.  C  e.  1o )
13 ssel 3187 . . . . . . 7  |-  ( ( A  ^o  B ) 
C_  1o  ->  ( C  e.  ( A  ^o  B )  ->  C  e.  1o ) )
145, 13syl5com 26 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  B )  C_  1o  ->  C  e.  1o ) )
1512, 14mtod 168 . . . . 5  |-  ( ph  ->  -.  ( A  ^o  B )  C_  1o )
16 difss 3316 . . . . . . . 8  |-  ( 1o 
\  B )  C_  1o
17 oe0m 6533 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( (/) 
^o  B )  =  ( 1o  \  B
) )
182, 17syl 15 . . . . . . . . 9  |-  ( ph  ->  ( (/)  ^o  B )  =  ( 1o  \  B ) )
1918sseq1d 3218 . . . . . . . 8  |-  ( ph  ->  ( ( (/)  ^o  B
)  C_  1o  <->  ( 1o  \  B )  C_  1o ) )
2016, 19mpbiri 224 . . . . . . 7  |-  ( ph  ->  ( (/)  ^o  B ) 
C_  1o )
21 oveq1 5881 . . . . . . . 8  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
2221sseq1d 3218 . . . . . . 7  |-  ( A  =  (/)  ->  ( ( A  ^o  B ) 
C_  1o  <->  ( (/)  ^o  B
)  C_  1o )
)
2320, 22syl5ibrcom 213 . . . . . 6  |-  ( ph  ->  ( A  =  (/)  ->  ( A  ^o  B
)  C_  1o )
)
24 oe1m 6559 . . . . . . . 8  |-  ( B  e.  On  ->  ( 1o  ^o  B )  =  1o )
25 eqimss 3243 . . . . . . . 8  |-  ( ( 1o  ^o  B )  =  1o  ->  ( 1o  ^o  B )  C_  1o )
262, 24, 253syl 18 . . . . . . 7  |-  ( ph  ->  ( 1o  ^o  B
)  C_  1o )
27 oveq1 5881 . . . . . . . 8  |-  ( A  =  1o  ->  ( A  ^o  B )  =  ( 1o  ^o  B
) )
2827sseq1d 3218 . . . . . . 7  |-  ( A  =  1o  ->  (
( A  ^o  B
)  C_  1o  <->  ( 1o  ^o  B )  C_  1o ) )
2926, 28syl5ibrcom 213 . . . . . 6  |-  ( ph  ->  ( A  =  1o 
->  ( A  ^o  B
)  C_  1o )
)
3023, 29jaod 369 . . . . 5  |-  ( ph  ->  ( ( A  =  (/)  \/  A  =  1o )  ->  ( A  ^o  B )  C_  1o ) )
3115, 30mtod 168 . . . 4  |-  ( ph  ->  -.  ( A  =  (/)  \/  A  =  1o ) )
32 elpri 3673 . . . . 5  |-  ( A  e.  { (/) ,  1o }  ->  ( A  =  (/)  \/  A  =  1o ) )
33 df2o3 6508 . . . . 5  |-  2o  =  { (/) ,  1o }
3432, 33eleq2s 2388 . . . 4  |-  ( A  e.  2o  ->  ( A  =  (/)  \/  A  =  1o ) )
3531, 34nsyl 113 . . 3  |-  ( ph  ->  -.  A  e.  2o )
36 eldif 3175 . . 3  |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  -.  A  e.  2o ) )
371, 35, 36sylanbrc 645 . 2  |-  ( ph  ->  A  e.  ( On 
\  2o ) )
3837, 10jca 518 1  |-  ( ph  ->  ( A  e.  ( On  \  2o )  /\  C  e.  ( On  \  1o ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    \ cdif 3162    C_ wss 3165   (/)c0 3468   {cpr 3654   {copab 4092   Oncon0 4408   dom cdm 4705   ran crn 4706   ` cfv 5271  (class class class)co 5874   1oc1o 6488   2oc2o 6489    ^o coe 6494   CNF ccnf 7378
This theorem is referenced by:  cantnflem3  7409  cantnflem4  7410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-oexp 6501
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