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Theorem undom 7602
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 7519 . . . . . . 7  |-  Rel  ~<_
21brrelex2i 5040 . . . . . 6  |-  ( A  ~<_  B  ->  B  e.  _V )
3 domeng 7527 . . . . . 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 241 . . . 4  |-  ( A  ~<_  B  ->  E. x
( A  ~~  x  /\  x  C_  B ) )
61brrelexi 5039 . . . . . . 7  |-  ( C  ~<_  D  ->  C  e.  _V )
7 difss 3631 . . . . . . 7  |-  ( C 
\  A )  C_  C
8 ssdomg 7558 . . . . . . 7  |-  ( C  e.  _V  ->  (
( C  \  A
)  C_  C  ->  ( C  \  A )  ~<_  C ) )
96, 7, 8mpisyl 18 . . . . . 6  |-  ( C  ~<_  D  ->  ( C  \  A )  ~<_  C )
10 domtr 7565 . . . . . 6  |-  ( ( ( C  \  A
)  ~<_  C  /\  C  ~<_  D )  ->  ( C  \  A )  ~<_  D )
119, 10mpancom 669 . . . . 5  |-  ( C  ~<_  D  ->  ( C  \  A )  ~<_  D )
121brrelex2i 5040 . . . . . . 7  |-  ( ( C  \  A )  ~<_  D  ->  D  e.  _V )
13 domeng 7527 . . . . . . 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 241 . . . . 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 566 . . 3  |-  ( ( A  ~<_  B  /\  C  ~<_  D )  ->  ( E. x ( A  ~~  x  /\  x  C_  B
)  /\  E. y
( ( C  \  A )  ~~  y  /\  y  C_  D ) ) )
1817adantr 465 . 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 1957 . . 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 761 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  ->  A  ~~  x )
21 simprrl 763 . . . . . . 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 3899 . . . . . . . 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 3725 . . . . . . . . . 10  |-  ( ( x  C_  B  /\  y  C_  D )  -> 
( x  i^i  y
)  C_  ( B  i^i  D ) )
2524ad2ant2l 745 . . . . . . . . 9  |-  ( ( ( A  ~~  x  /\  x  C_  B )  /\  ( ( C 
\  A )  ~~  y  /\  y  C_  D
) )  ->  (
x  i^i  y )  C_  ( B  i^i  D
) )
2625adantl 466 . . . . . . . 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 754 . . . . . . . 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 3817 . . . . . . . 8  |-  ( ( ( x  i^i  y
)  C_  ( B  i^i  D )  /\  ( B  i^i  D )  =  (/) )  ->  ( x  i^i  y )  =  (/) )
2926, 27, 28syl2anc 661 . . . . . . 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 3903 . . . . . . . 8  |-  ( A  u.  ( C  \  A ) )  =  ( A  u.  C
)
31 unen 7595 . . . . . . . 8  |-  ( ( ( A  ~~  x  /\  ( C  \  A
)  ~~  y )  /\  ( ( A  i^i  ( C  \  A ) )  =  (/)  /\  (
x  i^i  y )  =  (/) ) )  -> 
( A  u.  ( C  \  A ) ) 
~~  ( x  u.  y ) )
3230, 31syl5eqbrr 4481 . . . . . . 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 1229 . . . . . 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 729 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  ->  B  e.  _V )
351brrelex2i 5040 . . . . . . . . 9  |-  ( C  ~<_  D  ->  D  e.  _V )
3635ad3antlr 730 . . . . . . . 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 6583 . . . . . . . 8  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  ( B  u.  D
)  e.  _V )
3834, 36, 37syl2anc 661 . . . . . . 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 3676 . . . . . . . . 9  |-  ( ( x  C_  B  /\  y  C_  D )  -> 
( x  u.  y
)  C_  ( B  u.  D ) )
4039ad2ant2l 745 . . . . . . . 8  |-  ( ( ( A  ~~  x  /\  x  C_  B )  /\  ( ( C 
\  A )  ~~  y  /\  y  C_  D
) )  ->  (
x  u.  y ) 
C_  ( B  u.  D ) )
4140adantl 466 . . . . . . 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 7558 . . . . . . 7  |-  ( ( B  u.  D )  e.  _V  ->  (
( x  u.  y
)  C_  ( B  u.  D )  ->  (
x  u.  y )  ~<_  ( B  u.  D
) ) )
4338, 41, 42sylc 60 . . . . . 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 7570 . . . . . 6  |-  ( ( ( A  u.  C
)  ~~  ( x  u.  y )  /\  (
x  u.  y )  ~<_  ( B  u.  D
) )  ->  ( A  u.  C )  ~<_  ( B  u.  D
) )
4533, 43, 44syl2anc 661 . . . . 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 434 . . . 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 1701 . . 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 218 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   class class class wbr 4447    ~~ cen 7510    ~<_ cdom 7511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-en 7514  df-dom 7515
This theorem is referenced by:  domunsncan  7614  domunsn  7664  sucdom2  7711  unxpdom2  7725  sucxpdom  7726  fodomfi  7795  uncdadom  8547  cdadom1  8562
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