Theorem phplem3 7145

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 7147. (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 4543	. 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 7144	. . 3 |- ( ( A e. om /\ B e. A ) -> A ~~ ( suc A \ { B } ) )
5	2	enref 7041	. . . 4 |- A ~~ A
6		nnord 4754	. . . . . 6 |- ( A e. om -> Ord A )
7		orddif 4689	. . . . . 6 |- ( Ord A -> A = ( suc A \ { A } ) )
8	6, 7	syl 14	. . . . 5 |- ( A e. om -> A = ( suc A \ { A } ) )
9		sneq 3716	. . . . . . 7 |- ( A = B -> { A } = { B } )
10	9	difeq2d 3347	. . . . . 6 |- ( A = B -> ( suc A \ { A } ) = ( suc A \ { B } ) )
11	10	eqcoms 2241	. . . . 5 |- ( B = A -> ( suc A \ { A } ) = ( suc A \ { B } ) )
12	8, 11	sylan9eq 2291	. . . 4 |- ( ( A e. om /\ B = A ) -> A = ( suc A \ { B } ) )
13	5, 12	breqtrid 4162	. . 3 |- ( ( A e. om /\ B = A ) -> A ~~ ( suc A \ { B } ) )
14	4, 13	jaodan 809	. 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 720   = wceq 1402   e. wcel 2209   _Vcvv 2821   \ cdif 3217   {csn 3705   class class class wbr 4125   Ord word 4502   suc csuc 4505   omcom 4732   ~~ cen 7010

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-en 7013

This theorem is referenced by:  phplem4  7146  phplem3g  7147