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Theorem ordunidif 4878
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 4854 . . . . . . . 8  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  On )
2 onelss 4872 . . . . . . . 8  |-  ( B  e.  On  ->  (
x  e.  B  ->  x  C_  B ) )
31, 2syl 16 . . . . . . 7  |-  ( ( Ord  A  /\  B  e.  A )  ->  (
x  e.  B  ->  x  C_  B ) )
4 eloni 4840 . . . . . . . . . . 11  |-  ( B  e.  On  ->  Ord  B )
5 ordirr 4848 . . . . . . . . . . 11  |-  ( Ord 
B  ->  -.  B  e.  B )
64, 5syl 16 . . . . . . . . . 10  |-  ( B  e.  On  ->  -.  B  e.  B )
7 eldif 3449 . . . . . . . . . . 11  |-  ( B  e.  ( A  \  B )  <->  ( B  e.  A  /\  -.  B  e.  B ) )
87simplbi2 625 . . . . . . . . . 10  |-  ( B  e.  A  ->  ( -.  B  e.  B  ->  B  e.  ( A 
\  B ) ) )
96, 8syl5 32 . . . . . . . . 9  |-  ( B  e.  A  ->  ( B  e.  On  ->  B  e.  ( A  \  B ) ) )
109adantl 466 . . . . . . . 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 543 . . . . . 6  |-  ( ( Ord  A  /\  B  e.  A )  ->  (
x  e.  B  -> 
( B  e.  ( A  \  B )  /\  x  C_  B
) ) )
1312adantr 465 . . . . 5  |-  ( ( ( Ord  A  /\  B  e.  A )  /\  x  e.  A
)  ->  ( x  e.  B  ->  ( B  e.  ( A  \  B )  /\  x  C_  B ) ) )
14 sseq2 3489 . . . . . 6  |-  ( y  =  B  ->  (
x  C_  y  <->  x  C_  B
) )
1514rspcev 3179 . . . . 5  |-  ( ( B  e.  ( A 
\  B )  /\  x  C_  B )  ->  E. y  e.  ( A  \  B ) x 
C_  y )
1613, 15syl6 33 . . . 4  |-  ( ( ( Ord  A  /\  B  e.  A )  /\  x  e.  A
)  ->  ( x  e.  B  ->  E. y  e.  ( A  \  B
) x  C_  y
) )
17 eldif 3449 . . . . . . . . 9  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
1817biimpri 206 . . . . . . . 8  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  ->  x  e.  ( A  \  B ) )
19 ssid 3486 . . . . . . . 8  |-  x  C_  x
2018, 19jctir 538 . . . . . . 7  |-  ( ( x  e.  A  /\  -.  x  e.  B
)  ->  ( x  e.  ( A  \  B
)  /\  x  C_  x
) )
2120ex 434 . . . . . 6  |-  ( x  e.  A  ->  ( -.  x  e.  B  ->  ( x  e.  ( A  \  B )  /\  x  C_  x
) ) )
22 sseq2 3489 . . . . . . 7  |-  ( y  =  x  ->  (
x  C_  y  <->  x  C_  x
) )
2322rspcev 3179 . . . . . 6  |-  ( ( x  e.  ( A 
\  B )  /\  x  C_  x )  ->  E. y  e.  ( A  \  B ) x 
C_  y )
2421, 23syl6 33 . . . . 5  |-  ( x  e.  A  ->  ( -.  x  e.  B  ->  E. y  e.  ( A  \  B ) x  C_  y )
)
2524adantl 466 . . . 4  |-  ( ( ( Ord  A  /\  B  e.  A )  /\  x  e.  A
)  ->  ( -.  x  e.  B  ->  E. y  e.  ( A 
\  B ) x 
C_  y ) )
2616, 25pm2.61d 158 . . 3  |-  ( ( ( Ord  A  /\  B  e.  A )  /\  x  e.  A
)  ->  E. y  e.  ( A  \  B
) x  C_  y
)
2726ralrimiva 2830 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  A. x  e.  A  E. y  e.  ( A  \  B
) x  C_  y
)
28 unidif 4236 . 2  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) x  C_  y  ->  U. ( A  \  B )  =  U. A )
2927, 28syl 16 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  U. ( A  \  B )  = 
U. A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   E.wrex 2800    \ cdif 3436    C_ wss 3439   U.cuni 4202   Ord word 4829   Oncon0 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834
This theorem is referenced by: (None)
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