Theorem php 8763

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 8758 through phplem4 8761, nneneq 8762, 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 4295	. . . . . . . 8 ⊢ ∅ ⊆ 𝐵
2		sspsstr 4006	. . . . . . . 8 ⊢ ((∅ ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐴) → ∅ ⊊ 𝐴)
3	1, 2	mpan 690	. . . . . . 7 ⊢ (𝐵 ⊊ 𝐴 → ∅ ⊊ 𝐴)
4		0pss 4344	. . . . . . . 8 ⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)
5		df-ne 2936	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
6	4, 5	bitri 278	. . . . . . 7 ⊢ (∅ ⊊ 𝐴 ↔ ¬ 𝐴 = ∅)
7	3, 6	sylib 221	. . . . . 6 ⊢ (𝐵 ⊊ 𝐴 → ¬ 𝐴 = ∅)
8		nn0suc 7638	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
9	8	orcanai 1002	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
10	7, 9	sylan2 596	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
11		pssnel 4370	. . . . . . . . . 10 ⊢ (𝐵 ⊊ suc 𝑥 → ∃𝑦(𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦 ∈ 𝐵))
12		pssss 3996	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ⊊ suc 𝑥 → 𝐵 ⊆ suc 𝑥)
13		ssdif 4040	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ⊆ suc 𝑥 → (𝐵 ∖ {𝑦}) ⊆ (suc 𝑥 ∖ {𝑦}))
14		disjsn 4612	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝐵)
15		disj3 4353	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∩ {𝑦}) = ∅ ↔ 𝐵 = (𝐵 ∖ {𝑦}))
16	14, 15	bitr3i 280	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑦 ∈ 𝐵 ↔ 𝐵 = (𝐵 ∖ {𝑦}))
17		sseq1 3912	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = (𝐵 ∖ {𝑦}) → (𝐵 ⊆ (suc 𝑥 ∖ {𝑦}) ↔ (𝐵 ∖ {𝑦}) ⊆ (suc 𝑥 ∖ {𝑦})))
18	16, 17	sylbi 220	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑦 ∈ 𝐵 → (𝐵 ⊆ (suc 𝑥 ∖ {𝑦}) ↔ (𝐵 ∖ {𝑦}) ⊆ (suc 𝑥 ∖ {𝑦})))
19	13, 18	syl5ibr 249	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑦 ∈ 𝐵 → (𝐵 ⊆ suc 𝑥 → 𝐵 ⊆ (suc 𝑥 ∖ {𝑦})))
20		vex 3404	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
21	20	sucex 7557	. . . . . . . . . . . . . . . . . . 19 ⊢ suc 𝑥 ∈ V
22	21	difexi 5206	. . . . . . . . . . . . . . . . . 18 ⊢ (suc 𝑥 ∖ {𝑦}) ∈ V
23		ssdomg 8613	. . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑦 ∈ 𝐵 ∧ 𝐵 ⊊ suc 𝑥) → 𝐵 ≼ (suc 𝑥 ∖ {𝑦}))
27		vex 3404	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
28	20, 27	phplem3 8760	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥) → 𝑥 ≈ (suc 𝑥 ∖ {𝑦}))
29	28	ensymd 8618	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥) → (suc 𝑥 ∖ {𝑦}) ≈ 𝑥)
30		domentr 8626	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ≼ (suc 𝑥 ∖ {𝑦}) ∧ (suc 𝑥 ∖ {𝑦}) ≈ 𝑥) → 𝐵 ≼ 𝑥)
31	26, 29, 30	syl2an 599	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑦 ∈ 𝐵 ∧ 𝐵 ⊊ suc 𝑥) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥)) → 𝐵 ≼ 𝑥)
32	31	exp43 440	. . . . . . . . . . . . 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 410	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦 ∈ 𝐵) → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵 ≼ 𝑥)))
35	34	exlimiv 1937	. . . . . . . . . 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 8625	. . . . . . . . . . . 12 ⊢ ((suc 𝑥 ≈ 𝐵 ∧ 𝐵 ≼ 𝑥) → suc 𝑥 ≼ 𝑥)
38		sssucid 6259	. . . . . . . . . . . . 13 ⊢ 𝑥 ⊆ suc 𝑥
39		ssdomg 8613	. . . . . . . . . . . . 13 ⊢ (suc 𝑥 ∈ V → (𝑥 ⊆ suc 𝑥 → 𝑥 ≼ suc 𝑥))
40	21, 38, 39	mp2 9	. . . . . . . . . . . 12 ⊢ 𝑥 ≼ suc 𝑥
41		sbth 8699	. . . . . . . . . . . 12 ⊢ ((suc 𝑥 ≼ 𝑥 ∧ 𝑥 ≼ suc 𝑥) → suc 𝑥 ≈ 𝑥)
42	37, 40, 41	sylancl 589	. . . . . . . . . . 11 ⊢ ((suc 𝑥 ≈ 𝐵 ∧ 𝐵 ≼ 𝑥) → suc 𝑥 ≈ 𝑥)
43	42	expcom 417	. . . . . . . . . 10 ⊢ (𝐵 ≼ 𝑥 → (suc 𝑥 ≈ 𝐵 → suc 𝑥 ≈ 𝑥))
44		peano2b 7627	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
45		nnord 7619	. . . . . . . . . . . . 13 ⊢ (suc 𝑥 ∈ ω → Ord suc 𝑥)
46	44, 45	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ω → Ord suc 𝑥)
47	20	sucid 6261	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ suc 𝑥
48		nordeq 6201	. . . . . . . . . . . 12 ⊢ ((Ord suc 𝑥 ∧ 𝑥 ∈ suc 𝑥) → suc 𝑥 ≠ 𝑥)
49	46, 47, 48	sylancl 589	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ω → suc 𝑥 ≠ 𝑥)
50		nneneq 8762	. . . . . . . . . . . . . 14 ⊢ ((suc 𝑥 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 = 𝑥))
51	44, 50	sylanb 584	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 = 𝑥))
52	51	anidms 570	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ω → (suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 = 𝑥))
53	52	necon3bbid 2972	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ω → (¬ suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 ≠ 𝑥))
54	49, 53	mpbird 260	. . . . . . . . . 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 3989	. . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → (𝐵 ⊊ 𝐴 ↔ 𝐵 ⊊ suc 𝑥))
59		breq1 5043	. . . . . . . . 9 ⊢ (𝐴 = suc 𝑥 → (𝐴 ≈ 𝐵 ↔ suc 𝑥 ≈ 𝐵))
60	59	notbid 321	. . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → (¬ 𝐴 ≈ 𝐵 ↔ ¬ suc 𝑥 ≈ 𝐵))
61	58, 60	imbi12d 348	. . . . . . 7 ⊢ (𝐴 = suc 𝑥 → ((𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵) ↔ (𝐵 ⊊ suc 𝑥 → ¬ suc 𝑥 ≈ 𝐵)))
62	57, 61	syl5ibrcom 250	. . . . . 6 ⊢ (𝑥 ∈ ω → (𝐴 = suc 𝑥 → (𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵)))
63	62	rexlimiv 3191	. . . . 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 416	. . 3 ⊢ (𝐴 ∈ ω → (𝐵 ⊊ 𝐴 → (𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵)))
66	65	pm2.43d 53	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵))
67	66	imp 410	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐴 ≈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542  ∃wex 1786   ∈ wcel 2114   ≠ wne 2935  ∃wrex 3055  Vcvv 3400   ∖ cdif 3850   ∩ cin 3852   ⊆ wss 3853   ⊊ wpss 3854  ∅c0 4221  {csn 4526   class class class wbr 5040  Ord word 6181  suc csuc 6184  ωcom 7611   ≈ cen 8564   ≼ cdom 8565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-ord 6185  df-on 6186  df-lim 6187  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-om 7612  df-er 8332  df-en 8568  df-dom 8569

This theorem is referenced by:  php2  8764  php3  8765  rr-phpd  41408