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Theorem phplem1 7755
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 6712 . . 3  |-  ( A  e.  om  ->  Ord  A )
2 nordeq 5459 . . . 4  |-  ( ( Ord  A  /\  B  e.  A )  ->  A  =/=  B )
3 disjsn2 4059 . . . 4  |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
42, 3syl 17 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  ( { A }  i^i  { B } )  =  (/) )
51, 4sylan 474 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { A }  i^i  { B } )  =  (/) )
6 undif4 3850 . . 3  |-  ( ( { A }  i^i  { B } )  =  (/)  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( ( { A }  u.  A )  \  { B } ) )
7 df-suc 5446 . . . . 5  |-  suc  A  =  ( A  u.  { A } )
87equncomi 3613 . . . 4  |-  suc  A  =  ( { A }  u.  A )
98difeq1i 3580 . . 3  |-  ( suc 
A  \  { B } )  =  ( ( { A }  u.  A )  \  { B } )
106, 9syl6eqr 2482 . 2  |-  ( ( { A }  i^i  { B } )  =  (/)  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( suc 
A  \  { B } ) )
115, 10syl 17 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 371    = wceq 1438    e. wcel 1869    =/= wne 2619    \ cdif 3434    u. cun 3435    i^i cin 3436   (/)c0 3762   {csn 3997   Ord word 5439   suc csuc 5442   omcom 6704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-tr 4517  df-eprel 4762  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-om 6705
This theorem is referenced by:  phplem2  7756
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