Theorem phpm 6867

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 6854 through phplem4 6857, nneneq 6859, 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 3259	. . . . . . . . 9 |- ( x e. ( A \ B ) -> x e. A )
3		ne0i 3431	. . . . . . . . 9 |- ( x e. A -> A =/= (/) )
4	2, 3	syl 14	. . . . . . . 8 |- ( x e. ( A \ B ) -> A =/= (/) )
5	4	neneqd 2368	. . . . . . 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 620	. . . . 5 |- ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ A = (/) ) -> -. A ~~ B )
8		php5dom 6865	. . . . . . . . . 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 2742	. . . . . . . . . . . . . . . 16 |- y e. _V
13	12	sucex 4500	. . . . . . . . . . . . . . 15 |- suc y e. _V
14		difss 3263	. . . . . . . . . . . . . . 15 |- ( suc y \ { x } ) C_ suc y
15	13, 14	ssexi 4143	. . . . . . . . . . . . . 14 |- ( suc y \ { x } ) e. _V
16		eldifn 3260	. . . . . . . . . . . . . . . 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 3195	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> B C_ suc y )
22		ssdif 3272	. . . . . . . . . . . . . . . 16 |- ( B C_ suc y -> ( B \ { x } ) C_ ( suc y \ { x } ) )
23		disjsn 3656	. . . . . . . . . . . . . . . . . 18 |- ( ( B i^i { x } ) = (/) <-> -. x e. B )
24		disj3 3477	. . . . . . . . . . . . . . . . . 18 |- ( ( B i^i { x } ) = (/) <-> B = ( B \ { x } ) )
25	23, 24	bitr3i 186	. . . . . . . . . . . . . . . . 17 |- ( -. x e. B <-> B = ( B \ { x } ) )
26		sseq1 3180	. . . . . . . . . . . . . . . . 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 6780	. . . . . . . . . . . . . 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 2256	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> x e. suc y )
35		phplem3g 6858	. . . . . . . . . . . . . . 15 |- ( ( y e. om /\ x e. suc y ) -> y ~~ ( suc y \ { x } ) )
36	35	ensymd 6785	. . . . . . . . . . . . . 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 6793	. . . . . . . . . . . . 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 6792	. . . . . . . . . . 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 4029	. . . . . . . . 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 665	. . . . . . . 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 2594	. . . . . 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 4605	. . . . . 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 798	. . . 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 1819	. 2 |- ( ( A e. om /\ B C_ A ) -> ( E. x x e. ( A \ B ) -> -. A ~~ B ) )
53	52	3impia 1200	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 708   /\ w3a 978   = wceq 1353   E.wex 1492   e. wcel 2148   =/= wne 2347   E.wrex 2456   _Vcvv 2739   \ cdif 3128   i^i cin 3130   C_ wss 3131   (/)c0 3424   {csn 3594   class class class wbr 4005   suc csuc 4367   omcom 4591   ~~ cen 6740   ~<_ cdom 6741

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-er 6537  df-en 6743  df-dom 6744

This theorem is referenced by:  phpelm  6868