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Theorem cantnflem2 7579
Description: Lemma for cantnf 7582. (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 6717 . . . . . . . . . 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 4547 . . . . . . . . 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 6681 . . . . . . . 8  |-  ( C  e.  ( On  \  1o )  <->  ( C  e.  On  /\  (/)  e.  C
) )
107, 8, 9sylanbrc 646 . . . . . . 7  |-  ( ph  ->  C  e.  ( On 
\  1o ) )
1110eldifbd 3276 . . . . . 6  |-  ( ph  ->  -.  C  e.  1o )
12 ssel 3285 . . . . . . 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 6698 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
^o  B )  =  ( 1o  \  B
) )
162, 15syl 16 . . . . . . . 8  |-  ( ph  ->  ( (/)  ^o  B )  =  ( 1o  \  B ) )
17 difss 3417 . . . . . . . 8  |-  ( 1o 
\  B )  C_  1o
1816, 17syl6eqss 3341 . . . . . . 7  |-  ( ph  ->  ( (/)  ^o  B ) 
C_  1o )
19 oveq1 6027 . . . . . . . 8  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
2019sseq1d 3318 . . . . . . 7  |-  ( A  =  (/)  ->  ( ( A  ^o  B ) 
C_  1o  <->  ( (/)  ^o  B
)  C_  1o )
)
2118, 20syl5ibrcom 214 . . . . . 6  |-  ( ph  ->  ( A  =  (/)  ->  ( A  ^o  B
)  C_  1o )
)
22 oe1m 6724 . . . . . . . 8  |-  ( B  e.  On  ->  ( 1o  ^o  B )  =  1o )
23 eqimss 3343 . . . . . . . 8  |-  ( ( 1o  ^o  B )  =  1o  ->  ( 1o  ^o  B )  C_  1o )
242, 22, 233syl 19 . . . . . . 7  |-  ( ph  ->  ( 1o  ^o  B
)  C_  1o )
25 oveq1 6027 . . . . . . . 8  |-  ( A  =  1o  ->  ( A  ^o  B )  =  ( 1o  ^o  B
) )
2625sseq1d 3318 . . . . . . 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 3777 . . . . 5  |-  ( A  e.  { (/) ,  1o }  ->  ( A  =  (/)  \/  A  =  1o ) )
31 df2o3 6673 . . . . 5  |-  2o  =  { (/) ,  1o }
3230, 31eleq2s 2479 . . . 4  |-  ( A  e.  2o  ->  ( A  =  (/)  \/  A  =  1o ) )
3329, 32nsyl 115 . . 3  |-  ( ph  ->  -.  A  e.  2o )
341, 33eldifd 3274 . 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 1649    e. wcel 1717   A.wral 2649   E.wrex 2650    \ cdif 3260    C_ wss 3263   (/)c0 3571   {cpr 3758   {copab 4206   Oncon0 4522   dom cdm 4818   ran crn 4819   ` cfv 5394  (class class class)co 6020   1oc1o 6653   2oc2o 6654    ^o coe 6659   CNF ccnf 7549
This theorem is referenced by:  cantnflem3  7580  cantnflem4  7581
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-oexp 6666
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