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

Theorem ordunidif 5471
Description: The union of an ordinal stays the same if a subset equal to one of its elements is removed. (Contributed by NM, 10-Dec-2004.)
Assertion
Ref Expression
ordunidif  |-  ( ( Ord  A  /\  B  e.  A )  ->  U. ( A  \  B )  = 
U. A )

Proof of Theorem ordunidif
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordelon 5447 . . . . . . . 8  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  On )
2 onelss 5465 . . . . . . . 8  |-  ( B  e.  On  ->  (
x  e.  B  ->  x  C_  B ) )
31, 2syl 17 . . . . . . 7  |-  ( ( Ord  A  /\  B  e.  A )  ->  (
x  e.  B  ->  x  C_  B ) )
4 eloni 5433 . . . . . . . . . . 11  |-  ( B  e.  On  ->  Ord  B )
5 ordirr 5441 . . . . . . . . . . 11  |-  ( Ord 
B  ->  -.  B  e.  B )
64, 5syl 17 . . . . . . . . . 10  |-  ( B  e.  On  ->  -.  B  e.  B )
7 eldif 3414 . . . . . . . . . . 11  |-  ( B  e.  ( A  \  B )  <->  ( B  e.  A  /\  -.  B  e.  B ) )
87simplbi2 631 . . . . . . . . . 10  |-  ( B  e.  A  ->  ( -.  B  e.  B  ->  B  e.  ( A 
\  B ) ) )
96, 8syl5 33 . . . . . . . . 9  |-  ( B  e.  A  ->  ( B  e.  On  ->  B  e.  ( A  \  B ) ) )
109adantl 468 . . . . . . . 8  |-  ( ( Ord  A  /\  B  e.  A )  ->  ( B  e.  On  ->  B  e.  ( A  \  B ) ) )
111, 10mpd 15 . . . . . . 7  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  ( A  \  B
) )
123, 11jctild 546 . . . . . 6  |-  ( ( Ord  A  /\  B  e.  A )  ->  (
x  e.  B  -> 
( B  e.  ( A  \  B )  /\  x  C_  B
) ) )
1312adantr 467 . . . . 5  |-  ( ( ( Ord  A  /\  B  e.  A )  /\  x  e.  A
)  ->  ( x  e.  B  ->  ( B  e.  ( A  \  B )  /\  x  C_  B ) ) )
14 sseq2 3454 . . . . . 6  |-  ( y  =  B  ->  (
x  C_  y  <->  x  C_  B
) )
1514rspcev 3150 . . . . 5  |-  ( ( B  e.  ( A 
\  B )  /\  x  C_  B )  ->  E. y  e.  ( A  \  B ) x 
C_  y )
1613, 15syl6 34 . . . 4  |-  ( ( ( Ord  A  /\  B  e.  A )  /\  x  e.  A
)  ->  ( x  e.  B  ->  E. y  e.  ( A  \  B
) x  C_  y
) )
17 eldif 3414 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
1817biimpri 210 . . . . . . . 8  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  ->  x  e.  ( A  \  B ) )
19 ssid 3451 . . . . . . . 8  |-  x  C_  x
2018, 19jctir 541 . . . . . . 7  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  ->  ( x  e.  ( A  \  B
)  /\  x  C_  x
) )
2120ex 436 . . . . . 6  |-  ( x  e.  A  ->  ( -.  x  e.  B  ->  ( x  e.  ( A  \  B )  /\  x  C_  x
) ) )
22 sseq2 3454 . . . . . . 7  |-  ( y  =  x  ->  (
x  C_  y  <->  x  C_  x
) )
2322rspcev 3150 . . . . . 6  |-  ( ( x  e.  ( A 
\  B )  /\  x  C_  x )  ->  E. y  e.  ( A  \  B ) x 
C_  y )
2421, 23syl6 34 . . . . 5  |-  ( x  e.  A  ->  ( -.  x  e.  B  ->  E. y  e.  ( A  \  B ) x  C_  y )
)
2524adantl 468 . . . 4  |-  ( ( ( Ord  A  /\  B  e.  A )  /\  x  e.  A
)  ->  ( -.  x  e.  B  ->  E. y  e.  ( A 
\  B ) x 
C_  y ) )
2616, 25pm2.61d 162 . . 3  |-  ( ( ( Ord  A  /\  B  e.  A )  /\  x  e.  A
)  ->  E. y  e.  ( A  \  B
) x  C_  y
)
2726ralrimiva 2802 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  A. x  e.  A  E. y  e.  ( A  \  B
) x  C_  y
)
28 unidif 4231 . 2  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  U. ( A  \  B )  =  U. A )
2927, 28syl 17 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  U. ( A  \  B )  = 
U. A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   E.wrex 2738    \ cdif 3401    C_ wss 3404   U.cuni 4198   Ord word 5422   Oncon0 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-tr 4498  df-eprel 4745  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-ord 5426  df-on 5427
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator