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Theorem undom 6946
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 6865 . . . . . . 7  |-  Rel  ~<_
21brrelex2i 4729 . . . . . 6  |-  ( A  ~<_  B  ->  B  e.  _V )
3 domeng 6872 . . . . . 6  |-  ( B  e.  _V  ->  ( A  ~<_  B  <->  E. x
( A  ~~  x  /\  x  C_  B ) ) )
42, 3syl 15 . . . . 5  |-  ( A  ~<_  B  ->  ( A  ~<_  B 
<->  E. x ( A 
~~  x  /\  x  C_  B ) ) )
54ibi 232 . . . 4  |-  ( A  ~<_  B  ->  E. x
( A  ~~  x  /\  x  C_  B ) )
61brrelexi 4728 . . . . . . 7  |-  ( C  ~<_  D  ->  C  e.  _V )
7 difss 3304 . . . . . . 7  |-  ( C 
\  A )  C_  C
8 ssdomg 6903 . . . . . . 7  |-  ( C  e.  _V  ->  (
( C  \  A
)  C_  C  ->  ( C  \  A )  ~<_  C ) )
96, 7, 8ee10 1366 . . . . . 6  |-  ( C  ~<_  D  ->  ( C  \  A )  ~<_  C )
10 domtr 6910 . . . . . 6  |-  ( ( ( C  \  A
)  ~<_  C  /\  C  ~<_  D )  ->  ( C  \  A )  ~<_  D )
119, 10mpancom 650 . . . . 5  |-  ( C  ~<_  D  ->  ( C  \  A )  ~<_  D )
121brrelex2i 4729 . . . . . . 7  |-  ( ( C  \  A )  ~<_  D  ->  D  e.  _V )
13 domeng 6872 . . . . . . 7  |-  ( D  e.  _V  ->  (
( C  \  A
)  ~<_  D  <->  E. y
( ( C  \  A )  ~~  y  /\  y  C_  D ) ) )
1412, 13syl 15 . . . . . 6  |-  ( ( C  \  A )  ~<_  D  ->  ( ( C  \  A )  ~<_  D  <->  E. y ( ( C 
\  A )  ~~  y  /\  y  C_  D
) ) )
1514ibi 232 . . . . 5  |-  ( ( C  \  A )  ~<_  D  ->  E. y
( ( C  \  A )  ~~  y  /\  y  C_  D ) )
1611, 15syl 15 . . . 4  |-  ( C  ~<_  D  ->  E. y
( ( C  \  A )  ~~  y  /\  y  C_  D ) )
175, 16anim12i 549 . . 3  |-  ( ( A  ~<_  B  /\  C  ~<_  D )  ->  ( E. x ( A  ~~  x  /\  x  C_  B
)  /\  E. y
( ( C  \  A )  ~~  y  /\  y  C_  D ) ) )
1817adantr 451 . 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 1856 . . 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 738 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  ->  A  ~~  x )
21 simprrl 740 . . . . . . 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 3527 . . . . . . . 8  |-  ( A  i^i  ( C  \  A ) )  =  (/)
2322a1i 10 . . . . . . 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 3397 . . . . . . . . . 10  |-  ( ( x  C_  B  /\  y  C_  D )  -> 
( x  i^i  y
)  C_  ( B  i^i  D ) )
2524ad2ant2l 726 . . . . . . . . 9  |-  ( ( ( A  ~~  x  /\  x  C_  B )  /\  ( ( C 
\  A )  ~~  y  /\  y  C_  D
) )  ->  (
x  i^i  y )  C_  ( B  i^i  D
) )
2625adantl 452 . . . . . . . 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 731 . . . . . . . 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 3487 . . . . . . . 8  |-  ( ( ( x  i^i  y
)  C_  ( B  i^i  D )  /\  ( B  i^i  D )  =  (/) )  ->  ( x  i^i  y )  =  (/) )
2926, 27, 28syl2anc 642 . . . . . . 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 3531 . . . . . . . 8  |-  ( A  u.  ( C  \  A ) )  =  ( A  u.  C
)
31 unen 6939 . . . . . . . 8  |-  ( ( ( A  ~~  x  /\  ( C  \  A
)  ~~  y )  /\  ( ( A  i^i  ( C  \  A ) )  =  (/)  /\  (
x  i^i  y )  =  (/) ) )  -> 
( A  u.  ( C  \  A ) ) 
~~  ( x  u.  y ) )
3230, 31syl5eqbrr 4058 . . . . . . 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 1183 . . . . . 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 710 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  ->  B  e.  _V )
35 simpllr 735 . . . . . . . . 9  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  ->  C  ~<_  D )
361brrelex2i 4729 . . . . . . . . 9  |-  ( C  ~<_  D  ->  D  e.  _V )
3735, 36syl 15 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  ->  D  e.  _V )
38 unexg 4520 . . . . . . . 8  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  ( B  u.  D
)  e.  _V )
3934, 37, 38syl2anc 642 . . . . . . 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 )
40 unss12 3348 . . . . . . . . 9  |-  ( ( x  C_  B  /\  y  C_  D )  -> 
( x  u.  y
)  C_  ( B  u.  D ) )
4140ad2ant2l 726 . . . . . . . 8  |-  ( ( ( A  ~~  x  /\  x  C_  B )  /\  ( ( C 
\  A )  ~~  y  /\  y  C_  D
) )  ->  (
x  u.  y ) 
C_  ( B  u.  D ) )
4241adantl 452 . . . . . . 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 ) )
43 ssdomg 6903 . . . . . . 7  |-  ( ( B  u.  D )  e.  _V  ->  (
( x  u.  y
)  C_  ( B  u.  D )  ->  (
x  u.  y )  ~<_  ( B  u.  D
) ) )
4439, 42, 43sylc 56 . . . . . 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 ) )
45 endomtr 6915 . . . . . 6  |-  ( ( ( A  u.  C
)  ~~  ( x  u.  y )  /\  (
x  u.  y )  ~<_  ( B  u.  D
) )  ->  ( A  u.  C )  ~<_  ( B  u.  D
) )
4633, 44, 45syl2anc 642 . . . . 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 ) )
4746ex 423 . . . 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
) ) )
4847exlimdvv 1669 . . 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
) ) )
4919, 48syl5bir 209 . 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
) ) )
5018, 49mpd 14 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 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1685   _Vcvv 2789    \ cdif 3150    u. cun 3151    i^i cin 3152    C_ wss 3153   (/)c0 3456   class class class wbr 4024    ~~ cen 6856    ~<_ cdom 6857
This theorem is referenced by:  domunsncan  6958  domunsn  7007  sucdom2  7053  unxpdom2  7067  sucxpdom  7068  fodomfi  7131  uncdadom  7793  cdadom1  7808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-en 6860  df-dom 6861
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