Theorem php 8703

Description: Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. 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 8698 through phplem4 8701, nneneq 8702, and this final piece of the proof. (Contributed by NM, 29-May-1998.)

Assertion

Ref	Expression
php	⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐴 ≈ 𝐵)

Proof of Theorem php

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ss 4352	. . . . . . . 8 ⊢ ∅ ⊆ 𝐵
2		sspsstr 4084	. . . . . . . 8 ⊢ ((∅ ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐴) → ∅ ⊊ 𝐴)
3	1, 2	mpan 688	. . . . . . 7 ⊢ (𝐵 ⊊ 𝐴 → ∅ ⊊ 𝐴)
4		0pss 4398	. . . . . . . 8 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)
5		df-ne 3019	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
6	4, 5	bitri 277	. . . . . . 7 ⊢ (∅ ⊊ 𝐴 ↔ ¬ 𝐴 = ∅)
7	3, 6	sylib 220	. . . . . 6 ⊢ (𝐵 ⊊ 𝐴 → ¬ 𝐴 = ∅)
8		nn0suc 7608	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
9	8	orcanai 999	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
10	7, 9	sylan2 594	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
11		pssnel 4422	. . . . . . . . . 10 ⊢ (𝐵 ⊊ suc 𝑥 → ∃𝑦(𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦 ∈ 𝐵))
12		pssss 4074	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ⊊ suc 𝑥 → 𝐵 ⊆ suc 𝑥)
13		ssdif 4118	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ⊆ suc 𝑥 → (𝐵 ∖ {𝑦}) ⊆ (suc 𝑥 ∖ {𝑦}))
14		disjsn 4649	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝐵)
15		disj3 4405	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∩ {𝑦}) = ∅ ↔ 𝐵 = (𝐵 ∖ {𝑦}))
16	14, 15	bitr3i 279	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑦 ∈ 𝐵 ↔ 𝐵 = (𝐵 ∖ {𝑦}))
17		sseq1 3994	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = (𝐵 ∖ {𝑦}) → (𝐵 ⊆ (suc 𝑥 ∖ {𝑦}) ↔ (𝐵 ∖ {𝑦}) ⊆ (suc 𝑥 ∖ {𝑦})))
18	16, 17	sylbi 219	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑦 ∈ 𝐵 → (𝐵 ⊆ (suc 𝑥 ∖ {𝑦}) ↔ (𝐵 ∖ {𝑦}) ⊆ (suc 𝑥 ∖ {𝑦})))
19	13, 18	syl5ibr 248	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑦 ∈ 𝐵 → (𝐵 ⊆ suc 𝑥 → 𝐵 ⊆ (suc 𝑥 ∖ {𝑦})))
20		vex 3499	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
21	20	sucex 7528	. . . . . . . . . . . . . . . . . . 19 ⊢ suc 𝑥 ∈ V
22	21	difexi 5234	. . . . . . . . . . . . . . . . . 18 ⊢ (suc 𝑥 ∖ {𝑦}) ∈ V
23		ssdomg 8557	. . . . . . . . . . . . . . . . . 18 ⊢ ((suc 𝑥 ∖ {𝑦}) ∈ V → (𝐵 ⊆ (suc 𝑥 ∖ {𝑦}) → 𝐵 ≼ (suc 𝑥 ∖ {𝑦})))
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ⊆ (suc 𝑥 ∖ {𝑦}) → 𝐵 ≼ (suc 𝑥 ∖ {𝑦}))
25	12, 19, 24	syl56 36	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑦 ∈ 𝐵 → (𝐵 ⊊ suc 𝑥 → 𝐵 ≼ (suc 𝑥 ∖ {𝑦})))
26	25	imp 409	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑦 ∈ 𝐵 ∧ 𝐵 ⊊ suc 𝑥) → 𝐵 ≼ (suc 𝑥 ∖ {𝑦}))
27		vex 3499	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
28	20, 27	phplem3 8700	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥) → 𝑥 ≈ (suc 𝑥 ∖ {𝑦}))
29	28	ensymd 8562	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥) → (suc 𝑥 ∖ {𝑦}) ≈ 𝑥)
30		domentr 8570	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ≼ (suc 𝑥 ∖ {𝑦}) ∧ (suc 𝑥 ∖ {𝑦}) ≈ 𝑥) → 𝐵 ≼ 𝑥)
31	26, 29, 30	syl2an 597	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑦 ∈ 𝐵 ∧ 𝐵 ⊊ suc 𝑥) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥)) → 𝐵 ≼ 𝑥)
32	31	exp43 439	. . . . . . . . . . . . 13 ⊢ (¬ 𝑦 ∈ 𝐵 → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → (𝑦 ∈ suc 𝑥 → 𝐵 ≼ 𝑥))))
33	32	com4r 94	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ suc 𝑥 → (¬ 𝑦 ∈ 𝐵 → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵 ≼ 𝑥))))
34	33	imp 409	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦 ∈ 𝐵) → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵 ≼ 𝑥)))
35	34	exlimiv 1931	. . . . . . . . . 10 ⊢ (∃𝑦(𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦 ∈ 𝐵) → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵 ≼ 𝑥)))
36	11, 35	mpcom 38	. . . . . . . . 9 ⊢ (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵 ≼ 𝑥))
37		endomtr 8569	. . . . . . . . . . . 12 ⊢ ((suc 𝑥 ≈ 𝐵 ∧ 𝐵 ≼ 𝑥) → suc 𝑥 ≼ 𝑥)
38		sssucid 6270	. . . . . . . . . . . . 13 ⊢ 𝑥 ⊆ suc 𝑥
39		ssdomg 8557	. . . . . . . . . . . . 13 ⊢ (suc 𝑥 ∈ V → (𝑥 ⊆ suc 𝑥 → 𝑥 ≼ suc 𝑥))
40	21, 38, 39	mp2 9	. . . . . . . . . . . 12 ⊢ 𝑥 ≼ suc 𝑥
41		sbth 8639	. . . . . . . . . . . 12 ⊢ ((suc 𝑥 ≼ 𝑥 ∧ 𝑥 ≼ suc 𝑥) → suc 𝑥 ≈ 𝑥)
42	37, 40, 41	sylancl 588	. . . . . . . . . . 11 ⊢ ((suc 𝑥 ≈ 𝐵 ∧ 𝐵 ≼ 𝑥) → suc 𝑥 ≈ 𝑥)
43	42	expcom 416	. . . . . . . . . 10 ⊢ (𝐵 ≼ 𝑥 → (suc 𝑥 ≈ 𝐵 → suc 𝑥 ≈ 𝑥))
44		peano2b 7598	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
45		nnord 7590	. . . . . . . . . . . . 13 ⊢ (suc 𝑥 ∈ ω → Ord suc 𝑥)
46	44, 45	sylbi 219	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ω → Ord suc 𝑥)
47	20	sucid 6272	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ suc 𝑥
48		nordeq 6212	. . . . . . . . . . . 12 ⊢ ((Ord suc 𝑥 ∧ 𝑥 ∈ suc 𝑥) → suc 𝑥 ≠ 𝑥)
49	46, 47, 48	sylancl 588	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ω → suc 𝑥 ≠ 𝑥)
50		nneneq 8702	. . . . . . . . . . . . . 14 ⊢ ((suc 𝑥 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 = 𝑥))
51	44, 50	sylanb 583	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 = 𝑥))
52	51	anidms 569	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ω → (suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 = 𝑥))
53	52	necon3bbid 3055	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ω → (¬ suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 ≠ 𝑥))
54	49, 53	mpbird 259	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → ¬ suc 𝑥 ≈ 𝑥)
55	43, 54	nsyli 160	. . . . . . . . 9 ⊢ (𝐵 ≼ 𝑥 → (𝑥 ∈ ω → ¬ suc 𝑥 ≈ 𝐵))
56	36, 55	syli 39	. . . . . . . 8 ⊢ (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → ¬ suc 𝑥 ≈ 𝐵))
57	56	com12 32	. . . . . . 7 ⊢ (𝑥 ∈ ω → (𝐵 ⊊ suc 𝑥 → ¬ suc 𝑥 ≈ 𝐵))
58		psseq2 4067	. . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → (𝐵 ⊊ 𝐴 ↔ 𝐵 ⊊ suc 𝑥))
59		breq1 5071	. . . . . . . . 9 ⊢ (𝐴 = suc 𝑥 → (𝐴 ≈ 𝐵 ↔ suc 𝑥 ≈ 𝐵))
60	59	notbid 320	. . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → (¬ 𝐴 ≈ 𝐵 ↔ ¬ suc 𝑥 ≈ 𝐵))
61	58, 60	imbi12d 347	. . . . . . 7 ⊢ (𝐴 = suc 𝑥 → ((𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵) ↔ (𝐵 ⊊ suc 𝑥 → ¬ suc 𝑥 ≈ 𝐵)))
62	57, 61	syl5ibrcom 249	. . . . . 6 ⊢ (𝑥 ∈ ω → (𝐴 = suc 𝑥 → (𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵)))
63	62	rexlimiv 3282	. . . . 5 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → (𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵))
64	10, 63	syl 17	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → (𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵))
65	64	ex 415	. . 3 ⊢ (𝐴 ∈ ω → (𝐵 ⊊ 𝐴 → (𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵)))
66	65	pm2.43d 53	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵))
67	66	imp 409	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐴 ≈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3018  ∃wrex 3141  Vcvv 3496   ∖ cdif 3935   ∩ cin 3937   ⊆ wss 3938   ⊊ wpss 3939  ∅c0 4293  {csn 4569   class class class wbr 5068  Ord word 6192  suc csuc 6195  ωcom 7582   ≈ cen 8508   ≼ cdom 8509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583  df-er 8291  df-en 8512  df-dom 8513

This theorem is referenced by:  php2  8704  php3  8705  rr-phpd  40569