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Theorem phplem1 6925
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 4555 . . 3  |-  ( A  e.  om  ->  Ord  A )
2 nordeq 4304 . . . 4  |-  ( ( Ord  A  /\  B  e.  A )  ->  A  =/=  B )
3 disjsn2 3598 . . . 4  |-  ( A  =/=  B  ->  ( { A }  i^i  { B } )  =  (/) )
42, 3syl 17 . . 3  |-  ( ( Ord  A  /\  B  e.  A )  ->  ( { A }  i^i  { B } )  =  (/) )
51, 4sylan 459 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { A }  i^i  { B } )  =  (/) )
6 undif4 3418 . . 3  |-  ( ( { A }  i^i  { B } )  =  (/)  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( ( { A }  u.  A )  \  { B } ) )
7 df-suc 4291 . . . . 5  |-  suc  A  =  ( A  u.  { A } )
8 uncom 3229 . . . . 5  |-  ( A  u.  { A }
)  =  ( { A }  u.  A
)
97, 8eqtri 2273 . . . 4  |-  suc  A  =  ( { A }  u.  A )
109difeq1i 3207 . . 3  |-  ( suc 
A  \  { B } )  =  ( ( { A }  u.  A )  \  { B } )
116, 10syl6eqr 2303 . 2  |-  ( ( { A }  i^i  { B } )  =  (/)  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( suc 
A  \  { B } ) )
125, 11syl 17 1  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( suc  A  \  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412    \ cdif 3075    u. cun 3076    i^i cin 3077   (/)c0 3362   {csn 3544   Ord word 4284   suc csuc 4287   omcom 4547
This theorem is referenced by:  phplem2  6926
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548
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