Theorem phplem3 7039

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 7041. (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 4500	. 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 7038	. . 3 |- ( ( A e. om /\ B e. A ) -> A ~~ ( suc A \ { B } ) )
5	2	enref 6937	. . . 4 |- A ~~ A
6		nnord 4710	. . . . . 6 |- ( A e. om -> Ord A )
7		orddif 4645	. . . . . 6 |- ( Ord A -> A = ( suc A \ { A } ) )
8	6, 7	syl 14	. . . . 5 |- ( A e. om -> A = ( suc A \ { A } ) )
9		sneq 3680	. . . . . . 7 |- ( A = B -> { A } = { B } )
10	9	difeq2d 3325	. . . . . 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 4125	. . 3 |- ( ( A e. om /\ B = A ) -> A ~~ ( suc A \ { B } ) )
14	4, 13	jaodan 804	. 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 715   = wceq 1397   e. wcel 2202   _Vcvv 2802   \ cdif 3197   {csn 3669   class class class wbr 4088   Ord word 4459   suc csuc 4462   omcom 4688   ~~ cen 6906

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-en 6909

This theorem is referenced by:  phplem4  7040  phplem3g  7041