Theorem phpm 7171

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 7157 through phplem4 7160, nneneq 7162, 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 3351	. . . . . . . . 9 |- ( x e. ( A \ B ) -> x e. A )
3		ne0i 3528	. . . . . . . . 9 |- ( x e. A -> A =/= (/) )
4	2, 3	syl 14	. . . . . . . 8 |- ( x e. ( A \ B ) -> A =/= (/) )
5	4	neneqd 2441	. . . . . . 7 |- ( x e. ( A \ B ) -> -. A = (/) )
6	5	ad2antlr 493	. . . . . 6 |- ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ A = (/) ) -> -. A = (/) )
7	1, 6	pm2.21dd 629	. . . . 5 |- ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ A = (/) ) -> -. A ~~ B )
8		php5dom 7168	. . . . . . . . . 10 |- ( y e. om -> -. suc y ~<_ y )
9	8	ad2antlr 493	. . . . . . . . 9 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> -. suc y ~<_ y )
10		simplr 533	. . . . . . . . . 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 2824	. . . . . . . . . . . . . . . 16 |- y e. _V
13	12	sucex 4648	. . . . . . . . . . . . . . 15 |- suc y e. _V
14		difss 3355	. . . . . . . . . . . . . . 15 |- ( suc y \ { x } ) C_ suc y
15	13, 14	ssexi 4273	. . . . . . . . . . . . . 14 |- ( suc y \ { x } ) e. _V
16		eldifn 3352	. . . . . . . . . . . . . . . 16 |- ( x e. ( A \ B ) -> -. x e. B )
17	16	ad3antlr 497	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> -. x e. B )
18		simpllr 540	. . . . . . . . . . . . . . . . 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 3286	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> B C_ suc y )
22		ssdif 3364	. . . . . . . . . . . . . . . 16 |- ( B C_ suc y -> ( B \ { x } ) C_ ( suc y \ { x } ) )
23		disjsn 3771	. . . . . . . . . . . . . . . . . 18 |- ( ( B i^i { x } ) = (/) <-> -. x e. B )
24		disj3 3577	. . . . . . . . . . . . . . . . . 18 |- ( ( B i^i { x } ) = (/) <-> B = ( B \ { x } ) )
25	23, 24	bitr3i 186	. . . . . . . . . . . . . . . . 17 |- ( -. x e. B <-> B = ( B \ { x } ) )
26		sseq1 3271	. . . . . . . . . . . . . . . . 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 7069	. . . . . . . . . . . . . 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 533	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> y e. om )
33	2	ad3antlr 497	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> x e. A )
34	33, 20	eleqtrd 2317	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> x e. suc y )
35		phplem3g 7161	. . . . . . . . . . . . . . 15 |- ( ( y e. om /\ x e. suc y ) -> y ~~ ( suc y \ { x } ) )
36	35	ensymd 7074	. . . . . . . . . . . . . 14 |- ( ( y e. om /\ x e. suc y ) -> ( suc y \ { x } ) ~~ y )
37	32, 34, 36	syl2anc 415	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) /\ y e. om ) /\ A = suc y ) -> ( suc y \ { x } ) ~~ y )
38		domentr 7082	. . . . . . . . . . . . 13 |- ( ( B ~<_ ( suc y \ { x } ) /\ ( suc y \ { x } ) ~~ y ) -> B ~<_ y )
39	31, 37, 38	syl2anc 415	. . . . . . . . . . . 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 7081	. . . . . . . . . . 11 |- ( ( A ~~ B /\ B ~<_ y ) -> A ~<_ y )
42	11, 40, 41	syl2anc 415	. . . . . . . . . 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 4155	. . . . . . . . 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 675	. . . . . . . 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 2668	. . . . . 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 4753	. . . . . 6 |- ( A e. om -> ( A = (/) \/ E. y e. om A = suc y ) )
49	48	ad2antrr 492	. . . . 5 |- ( ( ( A e. om /\ B C_ A ) /\ x e. ( A \ B ) ) -> ( A = (/) \/ E. y e. om A = suc y ) )
50	7, 47, 49	mpjaodan 810	. . . 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 1872	. 2 |- ( ( A e. om /\ B C_ A ) -> ( E. x x e. ( A \ B ) -> -. A ~~ B ) )
53	52	3impia 1231	1 |- ( ( A e. om /\ B C_ A /\ E. x x e. ( A \ B ) ) -> -. A ~~ B )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   =/= wne 2420   E.wrex 2529   _Vcvv 2821   \ cdif 3217   i^i cin 3219   C_ wss 3220   (/)c0 3520   {csn 3709   class class class wbr 4131   suc csuc 4512   omcom 4739   ~~ cen 7024   ~<_ cdom 7025

This proof depends on 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 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof 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-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-er 6811  df-en 7027  df-dom 7028

This theorem is used by:  phpelm  7172