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Theorem phplem1 7715
Description: Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element. (Contributed by NM, 25-May-1998.)
Assertion
Ref Expression
phplem1  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( suc  A  \  { B } ) )

Proof of Theorem phplem1
StepHypRef Expression
1 nnord 6707 . . 3  |-  ( A  e.  om  ->  Ord  A )
2 nordeq 4906 . . . 4  |-  ( ( Ord  A  /\  B  e.  A )  ->  A  =/=  B )
3 disjsn2 4093 . . . 4  |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
42, 3syl 16 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  ( { A }  i^i  { B } )  =  (/) )
51, 4sylan 471 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { A }  i^i  { B } )  =  (/) )
6 undif4 3886 . . 3  |-  ( ( { A }  i^i  { B } )  =  (/)  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( ( { A }  u.  A )  \  { B } ) )
7 df-suc 4893 . . . . 5  |-  suc  A  =  ( A  u.  { A } )
87equncomi 3646 . . . 4  |-  suc  A  =  ( { A }  u.  A )
98difeq1i 3614 . . 3  |-  ( suc 
A  \  { B } )  =  ( ( { A }  u.  A )  \  { B } )
106, 9syl6eqr 2516 . 2  |-  ( ( { A }  i^i  { B } )  =  (/)  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( suc 
A  \  { B } ) )
115, 10syl 16 1  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( suc  A  \  { B } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652    \ cdif 3468    u. cun 3469    i^i cin 3470   (/)c0 3793   {csn 4032   Ord word 4886   suc csuc 4889   omcom 6699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-om 6700
This theorem is referenced by:  phplem2  7716
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