Theorem php 8144

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 8139 through phplem4 8142, nneneq 8143, 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 3972	. . . . . . . 8 ⊢ ∅ ⊆ 𝐵
2		sspsstr 3712	. . . . . . . 8 ⊢ ((∅ ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐴) → ∅ ⊊ 𝐴)
3	1, 2	mpan 706	. . . . . . 7 ⊢ (𝐵 ⊊ 𝐴 → ∅ ⊊ 𝐴)
4		0pss 4013	. . . . . . . 8 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)
5		df-ne 2795	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
6	4, 5	bitri 264	. . . . . . 7 ⊢ (∅ ⊊ 𝐴 ↔ ¬ 𝐴 = ∅)
7	3, 6	sylib 208	. . . . . 6 ⊢ (𝐵 ⊊ 𝐴 → ¬ 𝐴 = ∅)
8		nn0suc 7090	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
9	8	orcanai 952	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
10	7, 9	sylan2 491	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
11		pssnel 4039	. . . . . . . . . 10 ⊢ (𝐵 ⊊ suc 𝑥 → ∃𝑦(𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦 ∈ 𝐵))
12		pssss 3702	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ⊊ suc 𝑥 → 𝐵 ⊆ suc 𝑥)
13		ssdif 3745	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ⊆ suc 𝑥 → (𝐵 ∖ {𝑦}) ⊆ (suc 𝑥 ∖ {𝑦}))
14		disjsn 4246	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝐵)
15		disj3 4021	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∩ {𝑦}) = ∅ ↔ 𝐵 = (𝐵 ∖ {𝑦}))
16	14, 15	bitr3i 266	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑦 ∈ 𝐵 ↔ 𝐵 = (𝐵 ∖ {𝑦}))
17		sseq1 3626	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = (𝐵 ∖ {𝑦}) → (𝐵 ⊆ (suc 𝑥 ∖ {𝑦}) ↔ (𝐵 ∖ {𝑦}) ⊆ (suc 𝑥 ∖ {𝑦})))
18	16, 17	sylbi 207	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑦 ∈ 𝐵 → (𝐵 ⊆ (suc 𝑥 ∖ {𝑦}) ↔ (𝐵 ∖ {𝑦}) ⊆ (suc 𝑥 ∖ {𝑦})))
19	13, 18	syl5ibr 236	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑦 ∈ 𝐵 → (𝐵 ⊆ suc 𝑥 → 𝐵 ⊆ (suc 𝑥 ∖ {𝑦})))
20		vex 3203	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
21	20	sucex 7011	. . . . . . . . . . . . . . . . . . 19 ⊢ suc 𝑥 ∈ V
22		difss 3737	. . . . . . . . . . . . . . . . . . 19 ⊢ (suc 𝑥 ∖ {𝑦}) ⊆ suc 𝑥
23	21, 22	ssexi 4803	. . . . . . . . . . . . . . . . . 18 ⊢ (suc 𝑥 ∖ {𝑦}) ∈ V
24		ssdomg 8001	. . . . . . . . . . . . . . . . . 18 ⊢ ((suc 𝑥 ∖ {𝑦}) ∈ V → (𝐵 ⊆ (suc 𝑥 ∖ {𝑦}) → 𝐵 ≼ (suc 𝑥 ∖ {𝑦})))
25	23, 24	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ⊆ (suc 𝑥 ∖ {𝑦}) → 𝐵 ≼ (suc 𝑥 ∖ {𝑦}))
26	12, 19, 25	syl56 36	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑦 ∈ 𝐵 → (𝐵 ⊊ suc 𝑥 → 𝐵 ≼ (suc 𝑥 ∖ {𝑦})))
27	26	imp 445	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑦 ∈ 𝐵 ∧ 𝐵 ⊊ suc 𝑥) → 𝐵 ≼ (suc 𝑥 ∖ {𝑦}))
28		vex 3203	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
29	20, 28	phplem3 8141	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥) → 𝑥 ≈ (suc 𝑥 ∖ {𝑦}))
30	29	ensymd 8007	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥) → (suc 𝑥 ∖ {𝑦}) ≈ 𝑥)
31		domentr 8015	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ≼ (suc 𝑥 ∖ {𝑦}) ∧ (suc 𝑥 ∖ {𝑦}) ≈ 𝑥) → 𝐵 ≼ 𝑥)
32	27, 30, 31	syl2an 494	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑦 ∈ 𝐵 ∧ 𝐵 ⊊ suc 𝑥) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥)) → 𝐵 ≼ 𝑥)
33	32	exp43 640	. . . . . . . . . . . . 13 ⊢ (¬ 𝑦 ∈ 𝐵 → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → (𝑦 ∈ suc 𝑥 → 𝐵 ≼ 𝑥))))
34	33	com4r 94	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ suc 𝑥 → (¬ 𝑦 ∈ 𝐵 → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵 ≼ 𝑥))))
35	34	imp 445	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦 ∈ 𝐵) → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵 ≼ 𝑥)))
36	35	exlimiv 1858	. . . . . . . . . 10 ⊢ (∃𝑦(𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦 ∈ 𝐵) → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵 ≼ 𝑥)))
37	11, 36	mpcom 38	. . . . . . . . 9 ⊢ (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵 ≼ 𝑥))
38		endomtr 8014	. . . . . . . . . . . 12 ⊢ ((suc 𝑥 ≈ 𝐵 ∧ 𝐵 ≼ 𝑥) → suc 𝑥 ≼ 𝑥)
39		sssucid 5802	. . . . . . . . . . . . 13 ⊢ 𝑥 ⊆ suc 𝑥
40		ssdomg 8001	. . . . . . . . . . . . 13 ⊢ (suc 𝑥 ∈ V → (𝑥 ⊆ suc 𝑥 → 𝑥 ≼ suc 𝑥))
41	21, 39, 40	mp2 9	. . . . . . . . . . . 12 ⊢ 𝑥 ≼ suc 𝑥
42		sbth 8080	. . . . . . . . . . . 12 ⊢ ((suc 𝑥 ≼ 𝑥 ∧ 𝑥 ≼ suc 𝑥) → suc 𝑥 ≈ 𝑥)
43	38, 41, 42	sylancl 694	. . . . . . . . . . 11 ⊢ ((suc 𝑥 ≈ 𝐵 ∧ 𝐵 ≼ 𝑥) → suc 𝑥 ≈ 𝑥)
44	43	expcom 451	. . . . . . . . . 10 ⊢ (𝐵 ≼ 𝑥 → (suc 𝑥 ≈ 𝐵 → suc 𝑥 ≈ 𝑥))
45		peano2b 7081	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
46		nnord 7073	. . . . . . . . . . . . 13 ⊢ (suc 𝑥 ∈ ω → Ord suc 𝑥)
47	45, 46	sylbi 207	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ω → Ord suc 𝑥)
48	20	sucid 5804	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ suc 𝑥
49		nordeq 5742	. . . . . . . . . . . 12 ⊢ ((Ord suc 𝑥 ∧ 𝑥 ∈ suc 𝑥) → suc 𝑥 ≠ 𝑥)
50	47, 48, 49	sylancl 694	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ω → suc 𝑥 ≠ 𝑥)
51		nneneq 8143	. . . . . . . . . . . . . 14 ⊢ ((suc 𝑥 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 = 𝑥))
52	45, 51	sylanb 489	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 = 𝑥))
53	52	anidms 677	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ω → (suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 = 𝑥))
54	53	necon3bbid 2831	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ω → (¬ suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 ≠ 𝑥))
55	50, 54	mpbird 247	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → ¬ suc 𝑥 ≈ 𝑥)
56	44, 55	nsyli 155	. . . . . . . . 9 ⊢ (𝐵 ≼ 𝑥 → (𝑥 ∈ ω → ¬ suc 𝑥 ≈ 𝐵))
57	37, 56	syli 39	. . . . . . . 8 ⊢ (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → ¬ suc 𝑥 ≈ 𝐵))
58	57	com12 32	. . . . . . 7 ⊢ (𝑥 ∈ ω → (𝐵 ⊊ suc 𝑥 → ¬ suc 𝑥 ≈ 𝐵))
59		psseq2 3695	. . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → (𝐵 ⊊ 𝐴 ↔ 𝐵 ⊊ suc 𝑥))
60		breq1 4656	. . . . . . . . 9 ⊢ (𝐴 = suc 𝑥 → (𝐴 ≈ 𝐵 ↔ suc 𝑥 ≈ 𝐵))
61	60	notbid 308	. . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → (¬ 𝐴 ≈ 𝐵 ↔ ¬ suc 𝑥 ≈ 𝐵))
62	59, 61	imbi12d 334	. . . . . . 7 ⊢ (𝐴 = suc 𝑥 → ((𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵) ↔ (𝐵 ⊊ suc 𝑥 → ¬ suc 𝑥 ≈ 𝐵)))
63	58, 62	syl5ibrcom 237	. . . . . 6 ⊢ (𝑥 ∈ ω → (𝐴 = suc 𝑥 → (𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵)))
64	63	rexlimiv 3027	. . . . 5 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → (𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵))
65	10, 64	syl 17	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → (𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵))
66	65	ex 450	. . 3 ⊢ (𝐴 ∈ ω → (𝐵 ⊊ 𝐴 → (𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵)))
67	66	pm2.43d 53	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵))
68	67	imp 445	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐴 ≈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574   ⊊ wpss 3575  ∅c0 3915  {csn 4177   class class class wbr 4653  Ord word 5722  suc csuc 5725  ωcom 7065   ≈ cen 7952   ≼ cdom 7953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-er 7742  df-en 7956  df-dom 7957

This theorem is referenced by:  php2  8145  php3  8146