Theorem phplem3 7011

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 7013. (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 4493	. 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 7010	. . 3 |- ( ( A e. om /\ B e. A ) -> A ~~ ( suc A \ { B } ) )
5	2	enref 6914	. . . 4 |- A ~~ A
6		nnord 4703	. . . . . 6 |- ( A e. om -> Ord A )
7		orddif 4638	. . . . . 6 |- ( Ord A -> A = ( suc A \ { A } ) )
8	6, 7	syl 14	. . . . 5 |- ( A e. om -> A = ( suc A \ { A } ) )
9		sneq 3677	. . . . . . 7 |- ( A = B -> { A } = { B } )
10	9	difeq2d 3322	. . . . . 6 |- ( A = B -> ( suc A \ { A } ) = ( suc A \ { B } ) )
11	10	eqcoms 2232	. . . . 5 |- ( B = A -> ( suc A \ { A } ) = ( suc A \ { B } ) )
12	8, 11	sylan9eq 2282	. . . 4 |- ( ( A e. om /\ B = A ) -> A = ( suc A \ { B } ) )
13	5, 12	breqtrid 4119	. . 3 |- ( ( A e. om /\ B = A ) -> A ~~ ( suc A \ { B } ) )
14	4, 13	jaodan 802	. 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 713   = wceq 1395   e. wcel 2200   _Vcvv 2799   \ cdif 3194   {csn 3666   class class class wbr 4082   Ord word 4452   suc csuc 4455   omcom 4681   ~~ cen 6883

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-en 6886

This theorem is referenced by:  phplem4  7012  phplem3g  7013