Theorem phpm 7035

Description: Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. By "proper subset" here we mean that there is an element which is in the natural number and not in the subset, or in symbols E. x x e. ( A \ B ) (which is stronger than not being equal in the absence of excluded middle). Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 7021 through phplem4 7024, nneneq 7026, and this final piece of the proof. (Contributed by NM, 29-May-1998.)

Assertion

Ref	Expression
phpm	|- ( ( A e. om /\ B C_ A /\ E. x x e. ( A \ B ) ) -> -. A ~~ B )

Distinct variable groups:   x, A   x, B

Proof of Theorem phpm

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . 6 |- ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ A = (/) ) -> A = (/) )
2		eldifi 3326	. . . . . . . . 9 |- ( x e. ( A \ B ) -> x e. A )
3		ne0i 3498	. . . . . . . . 9 |- ( x e. A -> A =/= (/) )
4	2, 3	syl 14	. . . . . . . 8 |- ( x e. ( A \ B ) -> A =/= (/) )
5	4	neneqd 2421	. . . . . . 7 |- ( x e. ( A \ B ) -> -. A = (/) )
6	5	ad2antlr 489	. . . . . 6 |- ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ A = (/) ) -> -. A = (/) )
7	1, 6	pm2.21dd 623	. . . . 5 |- ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ A = (/) ) -> -. A ~~ B )
8		php5dom 7032	. . . . . . . . . 10 |- ( y e. om -> -. suc y ~<_ y )
9	8	ad2antlr 489	. . . . . . . . 9 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> -. suc y ~<_ y )
10		simplr 528	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) /\ A ~~ B ) -> A = suc y )
11		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) /\ A ~~ B ) -> A ~~ B )
12		vex 2802	. . . . . . . . . . . . . . . 16 |- y e. _V
13	12	sucex 4591	. . . . . . . . . . . . . . 15 |- suc y e. _V
14		difss 3330	. . . . . . . . . . . . . . 15 |- ( suc y \ { x } ) C_ suc y
15	13, 14	ssexi 4222	. . . . . . . . . . . . . 14 |- ( suc y \ { x } ) e. _V
16		eldifn 3327	. . . . . . . . . . . . . . . 16 |- ( x e. ( A \ B ) -> -. x e. B )
17	16	ad3antlr 493	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> -. x e. B )
18		simpllr 534	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) -> B C_ A )
19	18	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> B C_ A )
20		simpr 110	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> A = suc y )
21	19, 20	sseqtrd 3262	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> B C_ suc y )
22		ssdif 3339	. . . . . . . . . . . . . . . 16 |- ( B C_ suc y -> ( B \ { x } ) C_ ( suc y \ { x } ) )
23		disjsn 3728	. . . . . . . . . . . . . . . . . 18 |- ( ( B i^i { x } ) = (/) <-> -. x e. B )
24		disj3 3544	. . . . . . . . . . . . . . . . . 18 |- ( ( B i^i { x } ) = (/) <-> B = ( B \ { x } ) )
25	23, 24	bitr3i 186	. . . . . . . . . . . . . . . . 17 |- ( -. x e. B <-> B = ( B \ { x } ) )
26		sseq1 3247	. . . . . . . . . . . . . . . . 17 |- ( B = ( B \ { x } ) -> ( B C_ ( suc y \ { x } ) <-> ( B \ { x } ) C_ ( suc y \ { x } ) ) )
27	25, 26	sylbi 121	. . . . . . . . . . . . . . . 16 |- ( -. x e. B -> ( B C_ ( suc y \ { x } ) <-> ( B \ { x } ) C_ ( suc y \ { x } ) ) )
28	22, 27	imbitrrid 156	. . . . . . . . . . . . . . 15 |- ( -. x e. B -> ( B C_ suc y -> B C_ ( suc y \ { x } ) ) )
29	17, 21, 28	sylc 62	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> B C_ ( suc y \ { x } ) )
30		ssdomg 6938	. . . . . . . . . . . . . 14 |- ( ( suc y \ { x } ) e. _V -> ( B C_ ( suc y \ { x } ) -> B ~<_ ( suc y \ { x } ) ) )
31	15, 29, 30	mpsyl 65	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> B ~<_ ( suc y \ { x } ) )
32		simplr 528	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> y e. om )
33	2	ad3antlr 493	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> x e. A )
34	33, 20	eleqtrd 2308	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> x e. suc y )
35		phplem3g 7025	. . . . . . . . . . . . . . 15 |- ( ( y e. om /\ x e. suc y ) -> y ~~ ( suc y \ { x } ) )
36	35	ensymd 6943	. . . . . . . . . . . . . 14 |- ( ( y e. om /\ x e. suc y ) -> ( suc y \ { x } ) ~~ y )
37	32, 34, 36	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> ( suc y \ { x } ) ~~ y )
38		domentr 6951	. . . . . . . . . . . . 13 |- ( ( B ~<_ ( suc y \ { x } ) /\ ( suc y \ { x } ) ~~ y ) -> B ~<_ y )
39	31, 37, 38	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> B ~<_ y )
40	39	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) /\ A ~~ B ) -> B ~<_ y )
41		endomtr 6950	. . . . . . . . . . 11 |- ( ( A ~~ B /\ B ~<_ y ) -> A ~<_ y )
42	11, 40, 41	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) /\ A ~~ B ) -> A ~<_ y )
43	10, 42	eqbrtrrd 4107	. . . . . . . . 9 |- ( ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) /\ A ~~ B ) -> suc y ~<_ y )
44	9, 43	mtand 669	. . . . . . . 8 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> -. A ~~ B )
45	44	ex 115	. . . . . . 7 |- ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) -> ( A = suc y -> -. A ~~ B ) )
46	45	rexlimdva 2648	. . . . . 6 |- ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) -> ( E. y e. om A = suc y -> -. A ~~ B ) )
47	46	imp 124	. . . . 5 |- ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ E. y e. om A = suc y ) -> -. A ~~ B )
48		nn0suc 4696	. . . . . 6 |- ( A e. om -> ( A = (/) \/ E. y e. om A = suc y ) )
49	48	ad2antrr 488	. . . . 5 |- ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) -> ( A = (/) \/ E. y e. om A = suc y ) )
50	7, 47, 49	mpjaodan 803	. . . 4 |- ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) -> -. A ~~ B )
51	50	ex 115	. . 3 |- ( ( A e. om /\ B C_ A ) -> ( x e. ( A \ B ) -> -. A ~~ B ) )
52	51	exlimdv 1865	. 2 |- ( ( A e. om /\ B C_ A ) -> ( E. x x e. ( A \ B ) -> -. A ~~ B ) )
53	52	3impia 1224	1 |- ( ( A e. om /\ B C_ A /\ E. x x e. ( A \ B ) ) -> -. A ~~ B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   =/= wne 2400   E.wrex 2509   _Vcvv 2799   \ cdif 3194   i^i cin 3196   C_ wss 3197   (/)c0 3491   {csn 3666   class class class wbr 4083   suc csuc 4456   omcom 4682   ~~ cen 6893   ~<_ cdom 6894

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 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

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-sbc 3029  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 3889  df-int 3924  df-br 4084  df-opab 4146  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-er 6688  df-en 6896  df-dom 6897

This theorem is referenced by:  phpelm  7036