MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  undom Structured version   Unicode version

Theorem undom 7598
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 7515 . . . . . . 7  |-  Rel  ~<_
21brrelex2i 5030 . . . . . 6  |-  ( A  ~<_  B  ->  B  e.  _V )
3 domeng 7523 . . . . . 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 5029 . . . . . . 7  |-  ( C  ~<_  D  ->  C  e.  _V )
7 difss 3617 . . . . . . 7  |-  ( C 
\  A )  C_  C
8 ssdomg 7554 . . . . . . 7  |-  ( C  e.  _V  ->  (
( C  \  A
)  C_  C  ->  ( C  \  A )  ~<_  C ) )
96, 7, 8mpisyl 18 . . . . . 6  |-  ( C  ~<_  D  ->  ( C  \  A )  ~<_  C )
10 domtr 7561 . . . . . 6  |-  ( ( ( C  \  A
)  ~<_  C  /\  C  ~<_  D )  ->  ( C  \  A )  ~<_  D )
119, 10mpancom 667 . . . . 5  |-  ( C  ~<_  D  ->  ( C  \  A )  ~<_  D )
121brrelex2i 5030 . . . . . . 7  |-  ( ( C  \  A )  ~<_  D  ->  D  e.  _V )
13 domeng 7523 . . . . . . 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 564 . . 3  |-  ( ( A  ~<_  B  /\  C  ~<_  D )  ->  ( E. x ( A  ~~  x  /\  x  C_  B
)  /\  E. y
( ( C  \  A )  ~~  y  /\  y  C_  D ) ) )
1817adantr 463 . 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 1993 . . 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 3888 . . . . . . . 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 3711 . . . . . . . . . 10  |-  ( ( x  C_  B  /\  y  C_  D )  -> 
( x  i^i  y
)  C_  ( B  i^i  D ) )
2524ad2ant2l 743 . . . . . . . . 9  |-  ( ( ( A  ~~  x  /\  x  C_  B )  /\  ( ( C 
\  A )  ~~  y  /\  y  C_  D
) )  ->  (
x  i^i  y )  C_  ( B  i^i  D
) )
2625adantl 464 . . . . . . . 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 753 . . . . . . . 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 3816 . . . . . . . 8  |-  ( ( ( x  i^i  y
)  C_  ( B  i^i  D )  /\  ( B  i^i  D )  =  (/) )  ->  ( x  i^i  y )  =  (/) )
2926, 27, 28syl2anc 659 . . . . . . 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 3892 . . . . . . . 8  |-  ( A  u.  ( C  \  A ) )  =  ( A  u.  C
)
31 unen 7591 . . . . . . . 8  |-  ( ( ( A  ~~  x  /\  ( C  \  A
)  ~~  y )  /\  ( ( A  i^i  ( C  \  A ) )  =  (/)  /\  (
x  i^i  y )  =  (/) ) )  -> 
( A  u.  ( C  \  A ) ) 
~~  ( x  u.  y ) )
3230, 31syl5eqbrr 4473 . . . . . . 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 1227 . . . . . 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 727 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  ( B  i^i  D )  =  (/) )  /\  ( ( A  ~~  x  /\  x  C_  B
)  /\  ( ( C  \  A )  ~~  y  /\  y  C_  D
) ) )  ->  B  e.  _V )
351brrelex2i 5030 . . . . . . . . 9  |-  ( C  ~<_  D  ->  D  e.  _V )
3635ad3antlr 728 . . . . . . . 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 6574 . . . . . . . 8  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  ( B  u.  D
)  e.  _V )
3834, 36, 37syl2anc 659 . . . . . . 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 3662 . . . . . . . . 9  |-  ( ( x  C_  B  /\  y  C_  D )  -> 
( x  u.  y
)  C_  ( B  u.  D ) )
4039ad2ant2l 743 . . . . . . . 8  |-  ( ( ( A  ~~  x  /\  x  C_  B )  /\  ( ( C 
\  A )  ~~  y  /\  y  C_  D
) )  ->  (
x  u.  y ) 
C_  ( B  u.  D ) )
4140adantl 464 . . . . . . 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 7554 . . . . . . 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 7566 . . . . . 6  |-  ( ( ( A  u.  C
)  ~~  ( x  u.  y )  /\  (
x  u.  y )  ~<_  ( B  u.  D
) )  ->  ( A  u.  C )  ~<_  ( B  u.  D
) )
4533, 43, 44syl2anc 659 . . . . 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 432 . . . 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 1730 . . 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 367    = wceq 1398   E.wex 1617    e. wcel 1823   _Vcvv 3106    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   class class class wbr 4439    ~~ cen 7506    ~<_ cdom 7507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-en 7510  df-dom 7511
This theorem is referenced by:  domunsncan  7610  domunsn  7660  sucdom2  7707  unxpdom2  7721  sucxpdom  7722  fodomfi  7791  uncdadom  8542  cdadom1  8557
  Copyright terms: Public domain W3C validator