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Theorem undom 7133
Description: Dominance law for union. Proposition 4.24(a) of [Mendelson] p. 257. (Contributed by NM, 3-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
undom  |-  ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  ->  ( A  u.  C )  ~<_  ( B  u.  D
) )

Proof of Theorem undom
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldom 7052 . . . . . . 7  |-  Rel  ~<_
21brrelex2i 4860 . . . . . 6  |-  ( A  ~<_  B  ->  B  e.  _V )
3 domeng 7059 . . . . . 6  |-  ( B  e.  _V  ->  ( A  ~<_  B  <->  E. x
( A  ~~  x  /\  x  C_  B ) ) )
42, 3syl 16 . . . . 5  |-  ( A  ~<_  B  ->  ( A  ~<_  B 
<->  E. x ( A 
~~  x  /\  x  C_  B ) ) )
54ibi 233 . . . 4  |-  ( A  ~<_  B  ->  E. x
( A  ~~  x  /\  x  C_  B ) )
61brrelexi 4859 . . . . . . 7  |-  ( C  ~<_  D  ->  C  e.  _V )
7 difss 3418 . . . . . . 7  |-  ( C 
\  A )  C_  C
8 ssdomg 7090 . . . . . . 7  |-  ( C  e.  _V  ->  (
( C  \  A
)  C_  C  ->  ( C  \  A )  ~<_  C ) )
96, 7, 8ee10 1382 . . . . . 6  |-  ( C  ~<_  D  ->  ( C  \  A )  ~<_  C )
10 domtr 7097 . . . . . 6  |-  ( ( ( C  \  A
)  ~<_  C  /\  C  ~<_  D )  ->  ( C  \  A )  ~<_  D )
119, 10mpancom 651 . . . . 5  |-  ( C  ~<_  D  ->  ( C  \  A )  ~<_  D )
121brrelex2i 4860 . . . . . . 7  |-  ( ( C  \  A )  ~<_  D  ->  D  e.  _V )
13 domeng 7059 . . . . . . 7  |-  ( D  e.  _V  ->  (
( C  \  A
)  ~<_  D  <->  E. y
( ( C  \  A )  ~~  y  /\  y  C_  D ) ) )
1412, 13syl 16 . . . . . 6  |-  ( ( C  \  A )  ~<_  D  ->  ( ( C  \  A )  ~<_  D  <->  E. y ( ( C 
\  A )  ~~  y  /\  y  C_  D
) ) )
1514ibi 233 . . . . 5  |-  ( ( C  \  A )  ~<_  D  ->  E. y
( ( C  \  A )  ~~  y  /\  y  C_  D ) )
1611, 15syl 16 . . . 4  |-  ( C  ~<_  D  ->  E. y
( ( C  \  A )  ~~  y  /\  y  C_  D ) )
175, 16anim12i 550 . . 3  |-  ( ( A  ~<_  B  /\  C  ~<_  D )  ->  ( E. x ( A  ~~  x  /\  x  C_  B
)  /\  E. y
( ( C  \  A )  ~~  y  /\  y  C_  D ) ) )
1817adantr 452 . 2  |-  ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  ->  ( E. x ( A  ~~  x  /\  x  C_  B
)  /\  E. y
( ( C  \  A )  ~~  y  /\  y  C_  D ) ) )
19 eeanv 1926 . . 3  |-  ( E. x E. y ( ( A  ~~  x  /\  x  C_  B )  /\  ( ( C 
\  A )  ~~  y  /\  y  C_  D
) )  <->  ( E. x ( A  ~~  x  /\  x  C_  B
)  /\  E. y
( ( C  \  A )  ~~  y  /\  y  C_  D ) ) )
20 simprll 739 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  ->  A  ~~  x )
21 simprrl 741 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  -> 
( C  \  A
)  ~~  y )
22 disjdif 3644 . . . . . . . 8  |-  ( A  i^i  ( C  \  A ) )  =  (/)
2322a1i 11 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  -> 
( A  i^i  ( C  \  A ) )  =  (/) )
24 ss2in 3512 . . . . . . . . . 10  |-  ( ( x  C_  B  /\  y  C_  D )  -> 
( x  i^i  y
)  C_  ( B  i^i  D ) )
2524ad2ant2l 727 . . . . . . . . 9  |-  ( ( ( A  ~~  x  /\  x  C_  B )  /\  ( ( C 
\  A )  ~~  y  /\  y  C_  D
) )  ->  (
x  i^i  y )  C_  ( B  i^i  D
) )
2625adantl 453 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  -> 
( x  i^i  y
)  C_  ( B  i^i  D ) )
27 simplr 732 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  -> 
( B  i^i  D
)  =  (/) )
28 sseq0 3603 . . . . . . . 8  |-  ( ( ( x  i^i  y
)  C_  ( B  i^i  D )  /\  ( B  i^i  D )  =  (/) )  ->  ( x  i^i  y )  =  (/) )
2926, 27, 28syl2anc 643 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  -> 
( x  i^i  y
)  =  (/) )
30 undif2 3648 . . . . . . . 8  |-  ( A  u.  ( C  \  A ) )  =  ( A  u.  C
)
31 unen 7126 . . . . . . . 8  |-  ( ( ( A  ~~  x  /\  ( C  \  A
)  ~~  y )  /\  ( ( A  i^i  ( C  \  A ) )  =  (/)  /\  (
x  i^i  y )  =  (/) ) )  -> 
( A  u.  ( C  \  A ) ) 
~~  ( x  u.  y ) )
3230, 31syl5eqbrr 4188 . . . . . . 7  |-  ( ( ( A  ~~  x  /\  ( C  \  A
)  ~~  y )  /\  ( ( A  i^i  ( C  \  A ) )  =  (/)  /\  (
x  i^i  y )  =  (/) ) )  -> 
( A  u.  C
)  ~~  ( x  u.  y ) )
3320, 21, 23, 29, 32syl22anc 1185 . . . . . 6  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  -> 
( A  u.  C
)  ~~  ( x  u.  y ) )
342ad3antrrr 711 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  ->  B  e.  _V )
351brrelex2i 4860 . . . . . . . . 9  |-  ( C  ~<_  D  ->  D  e.  _V )
3635ad3antlr 712 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  ->  D  e.  _V )
37 unexg 4651 . . . . . . . 8  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  ( B  u.  D
)  e.  _V )
3834, 36, 37syl2anc 643 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  -> 
( B  u.  D
)  e.  _V )
39 unss12 3463 . . . . . . . . 9  |-  ( ( x  C_  B  /\  y  C_  D )  -> 
( x  u.  y
)  C_  ( B  u.  D ) )
4039ad2ant2l 727 . . . . . . . 8  |-  ( ( ( A  ~~  x  /\  x  C_  B )  /\  ( ( C 
\  A )  ~~  y  /\  y  C_  D
) )  ->  (
x  u.  y ) 
C_  ( B  u.  D ) )
4140adantl 453 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  -> 
( x  u.  y
)  C_  ( B  u.  D ) )
42 ssdomg 7090 . . . . . . 7  |-  ( ( B  u.  D )  e.  _V  ->  (
( x  u.  y
)  C_  ( B  u.  D )  ->  (
x  u.  y )  ~<_  ( B  u.  D
) ) )
4338, 41, 42sylc 58 . . . . . 6  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  -> 
( x  u.  y
)  ~<_  ( B  u.  D ) )
44 endomtr 7102 . . . . . 6  |-  ( ( ( A  u.  C
)  ~~  ( x  u.  y )  /\  (
x  u.  y )  ~<_  ( B  u.  D
) )  ->  ( A  u.  C )  ~<_  ( B  u.  D
) )
4533, 43, 44syl2anc 643 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  -> 
( A  u.  C
)  ~<_  ( B  u.  D ) )
4645ex 424 . . . 4  |-  ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  ->  (
( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) )  ->  ( A  u.  C )  ~<_  ( B  u.  D
) ) )
4746exlimdvv 1644 . . 3  |-  ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  ->  ( E. x E. y ( ( A  ~~  x  /\  x  C_  B )  /\  ( ( C 
\  A )  ~~  y  /\  y  C_  D
) )  ->  ( A  u.  C )  ~<_  ( B  u.  D
) ) )
4819, 47syl5bir 210 . 2  |-  ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  ->  (
( E. x ( A  ~~  x  /\  x  C_  B )  /\  E. y ( ( C 
\  A )  ~~  y  /\  y  C_  D
) )  ->  ( A  u.  C )  ~<_  ( B  u.  D
) ) )
4918, 48mpd 15 1  |-  ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  ->  ( A  u.  C )  ~<_  ( B  u.  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2900    \ cdif 3261    u. cun 3262    i^i cin 3263    C_ wss 3264   (/)c0 3572   class class class wbr 4154    ~~ cen 7043    ~<_ cdom 7044
This theorem is referenced by:  domunsncan  7145  domunsn  7194  sucdom2  7240  unxpdom2  7254  sucxpdom  7255  fodomfi  7322  uncdadom  7985  cdadom1  8000
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-en 7047  df-dom 7048
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