Theorem phplem3 7083

Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. For a version without the redundant hypotheses, see phplem3g 7085. (Contributed by NM, 26-May-1998.)

Hypotheses

Ref	Expression
phplem2.1	|- A e. _V
phplem2.2	|- B e. _V

Assertion

Ref	Expression
phplem3	|- ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) )

Proof of Theorem phplem3

Step	Hyp	Ref	Expression
1		elsuci 4506	. 2 |- ( B e. suc A -> ( B e. A \/ B = A ) )
2		phplem2.1	. . . 4 |- A e. _V
3		phplem2.2	. . . 4 |- B e. _V
4	2, 3	phplem2 7082	. . 3 |- ( ( A e. om /\ B e. A ) -> A ~~ ( suc A \ { B } ) )
5	2	enref 6981	. . . 4 |- A ~~ A
6		nnord 4716	. . . . . 6 |- ( A e. om -> Ord A )
7		orddif 4651	. . . . . 6 |- ( Ord A -> A = ( suc A \ { A } ) )
8	6, 7	syl 14	. . . . 5 |- ( A e. om -> A = ( suc A \ { A } ) )
9		sneq 3684	. . . . . . 7 |- ( A = B -> { A } = { B } )
10	9	difeq2d 3327	. . . . . 6 |- ( A = B -> ( suc A \ { A } ) = ( suc A \ { B } ) )
11	10	eqcoms 2234	. . . . 5 |- ( B = A -> ( suc A \ { A } ) = ( suc A \ { B } ) )
12	8, 11	sylan9eq 2284	. . . 4 |- ( ( A e. om /\ B = A ) -> A = ( suc A \ { B } ) )
13	5, 12	breqtrid 4130	. . 3 |- ( ( A e. om /\ B = A ) -> A ~~ ( suc A \ { B } ) )
14	4, 13	jaodan 805	. 2 |- ( ( A e. om /\ ( B e. A \/ B = A ) ) -> A ~~ ( suc A \ { B } ) )
15	1, 14	sylan2 286	1 |- ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2202   _Vcvv 2803   \ cdif 3198   {csn 3673   class class class wbr 4093   Ord word 4465   suc csuc 4468   omcom 4694   ~~ cen 6950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-en 6953

This theorem is referenced by:  phplem4  7084  phplem3g  7085