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Theorem undom 7188
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 7107 . . . . . . 7  |-  Rel  ~<_
21brrelex2i 4911 . . . . . 6  |-  ( A  ~<_  B  ->  B  e.  _V )
3 domeng 7114 . . . . . 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 4910 . . . . . . 7  |-  ( C  ~<_  D  ->  C  e.  _V )
7 difss 3466 . . . . . . 7  |-  ( C 
\  A )  C_  C
8 ssdomg 7145 . . . . . . 7  |-  ( C  e.  _V  ->  (
( C  \  A
)  C_  C  ->  ( C  \  A )  ~<_  C ) )
96, 7, 8ee10 1385 . . . . . 6  |-  ( C  ~<_  D  ->  ( C  \  A )  ~<_  C )
10 domtr 7152 . . . . . 6  |-  ( ( ( C  \  A
)  ~<_  C  /\  C  ~<_  D )  ->  ( C  \  A )  ~<_  D )
119, 10mpancom 651 . . . . 5  |-  ( C  ~<_  D  ->  ( C  \  A )  ~<_  D )
121brrelex2i 4911 . . . . . . 7  |-  ( ( C  \  A )  ~<_  D  ->  D  e.  _V )
13 domeng 7114 . . . . . . 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 1937 . . 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 3692 . . . . . . . 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 3560 . . . . . . . . . 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 3651 . . . . . . . 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 3696 . . . . . . . 8  |-  ( A  u.  ( C  \  A ) )  =  ( A  u.  C
)
31 unen 7181 . . . . . . . 8  |-  ( ( ( A  ~~  x  /\  ( C  \  A
)  ~~  y )  /\  ( ( A  i^i  ( C  \  A ) )  =  (/)  /\  (
x  i^i  y )  =  (/) ) )  -> 
( A  u.  ( C  \  A ) ) 
~~  ( x  u.  y ) )
3230, 31syl5eqbrr 4238 . . . . . . 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 4911 . . . . . . . . 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 4702 . . . . . . . 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 3511 . . . . . . . . 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 7145 . . . . . . 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 7157 . . . . . 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 1647 . . 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 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   class class class wbr 4204    ~~ cen 7098    ~<_ cdom 7099
This theorem is referenced by:  domunsncan  7200  domunsn  7249  sucdom2  7295  unxpdom2  7309  sucxpdom  7310  fodomfi  7377  uncdadom  8043  cdadom1  8058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-en 7102  df-dom 7103
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