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Theorem php 8089
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 8084 through phplem4 8087, nneneq 8088, and this final piece of the proof. (Contributed by NM, 29-May-1998.)
Assertion
Ref Expression
php ((𝐴 ∈ ω ∧ 𝐵𝐴) → ¬ 𝐴𝐵)

Proof of Theorem php
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ss 3949 . . . . . . . 8 ∅ ⊆ 𝐵
2 sspsstr 3695 . . . . . . . 8 ((∅ ⊆ 𝐵𝐵𝐴) → ∅ ⊊ 𝐴)
31, 2mpan 705 . . . . . . 7 (𝐵𝐴 → ∅ ⊊ 𝐴)
4 0pss 3990 . . . . . . . 8 (∅ ⊊ 𝐴𝐴 ≠ ∅)
5 df-ne 2797 . . . . . . . 8 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
64, 5bitri 264 . . . . . . 7 (∅ ⊊ 𝐴 ↔ ¬ 𝐴 = ∅)
73, 6sylib 208 . . . . . 6 (𝐵𝐴 → ¬ 𝐴 = ∅)
8 nn0suc 7038 . . . . . . 7 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
98orcanai 951 . . . . . 6 ((𝐴 ∈ ω ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
107, 9sylan2 491 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
11 pssnel 4016 . . . . . . . . . 10 (𝐵 ⊊ suc 𝑥 → ∃𝑦(𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦𝐵))
12 pssss 3685 . . . . . . . . . . . . . . . . 17 (𝐵 ⊊ suc 𝑥𝐵 ⊆ suc 𝑥)
13 ssdif 3728 . . . . . . . . . . . . . . . . . 18 (𝐵 ⊆ suc 𝑥 → (𝐵 ∖ {𝑦}) ⊆ (suc 𝑥 ∖ {𝑦}))
14 disjsn 4221 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝐵)
15 disj3 3998 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∩ {𝑦}) = ∅ ↔ 𝐵 = (𝐵 ∖ {𝑦}))
1614, 15bitr3i 266 . . . . . . . . . . . . . . . . . . 19 𝑦𝐵𝐵 = (𝐵 ∖ {𝑦}))
17 sseq1 3610 . . . . . . . . . . . . . . . . . . 19 (𝐵 = (𝐵 ∖ {𝑦}) → (𝐵 ⊆ (suc 𝑥 ∖ {𝑦}) ↔ (𝐵 ∖ {𝑦}) ⊆ (suc 𝑥 ∖ {𝑦})))
1816, 17sylbi 207 . . . . . . . . . . . . . . . . . 18 𝑦𝐵 → (𝐵 ⊆ (suc 𝑥 ∖ {𝑦}) ↔ (𝐵 ∖ {𝑦}) ⊆ (suc 𝑥 ∖ {𝑦})))
1913, 18syl5ibr 236 . . . . . . . . . . . . . . . . 17 𝑦𝐵 → (𝐵 ⊆ suc 𝑥𝐵 ⊆ (suc 𝑥 ∖ {𝑦})))
20 vex 3194 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ V
2120sucex 6959 . . . . . . . . . . . . . . . . . . 19 suc 𝑥 ∈ V
22 difss 3720 . . . . . . . . . . . . . . . . . . 19 (suc 𝑥 ∖ {𝑦}) ⊆ suc 𝑥
2321, 22ssexi 4768 . . . . . . . . . . . . . . . . . 18 (suc 𝑥 ∖ {𝑦}) ∈ V
24 ssdomg 7946 . . . . . . . . . . . . . . . . . 18 ((suc 𝑥 ∖ {𝑦}) ∈ V → (𝐵 ⊆ (suc 𝑥 ∖ {𝑦}) → 𝐵 ≼ (suc 𝑥 ∖ {𝑦})))
2523, 24ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝐵 ⊆ (suc 𝑥 ∖ {𝑦}) → 𝐵 ≼ (suc 𝑥 ∖ {𝑦}))
2612, 19, 25syl56 36 . . . . . . . . . . . . . . . 16 𝑦𝐵 → (𝐵 ⊊ suc 𝑥𝐵 ≼ (suc 𝑥 ∖ {𝑦})))
2726imp 445 . . . . . . . . . . . . . . 15 ((¬ 𝑦𝐵𝐵 ⊊ suc 𝑥) → 𝐵 ≼ (suc 𝑥 ∖ {𝑦}))
28 vex 3194 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
2920, 28phplem3 8086 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥) → 𝑥 ≈ (suc 𝑥 ∖ {𝑦}))
3029ensymd 7952 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥) → (suc 𝑥 ∖ {𝑦}) ≈ 𝑥)
31 domentr 7960 . . . . . . . . . . . . . . 15 ((𝐵 ≼ (suc 𝑥 ∖ {𝑦}) ∧ (suc 𝑥 ∖ {𝑦}) ≈ 𝑥) → 𝐵𝑥)
3227, 30, 31syl2an 494 . . . . . . . . . . . . . 14 (((¬ 𝑦𝐵𝐵 ⊊ suc 𝑥) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥)) → 𝐵𝑥)
3332exp43 639 . . . . . . . . . . . . 13 𝑦𝐵 → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → (𝑦 ∈ suc 𝑥𝐵𝑥))))
3433com4r 94 . . . . . . . . . . . 12 (𝑦 ∈ suc 𝑥 → (¬ 𝑦𝐵 → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵𝑥))))
3534imp 445 . . . . . . . . . . 11 ((𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦𝐵) → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵𝑥)))
3635exlimiv 1860 . . . . . . . . . 10 (∃𝑦(𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦𝐵) → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵𝑥)))
3711, 36mpcom 38 . . . . . . . . 9 (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵𝑥))
38 endomtr 7959 . . . . . . . . . . . 12 ((suc 𝑥𝐵𝐵𝑥) → suc 𝑥𝑥)
39 sssucid 5764 . . . . . . . . . . . . 13 𝑥 ⊆ suc 𝑥
40 ssdomg 7946 . . . . . . . . . . . . 13 (suc 𝑥 ∈ V → (𝑥 ⊆ suc 𝑥𝑥 ≼ suc 𝑥))
4121, 39, 40mp2 9 . . . . . . . . . . . 12 𝑥 ≼ suc 𝑥
42 sbth 8025 . . . . . . . . . . . 12 ((suc 𝑥𝑥𝑥 ≼ suc 𝑥) → suc 𝑥𝑥)
4338, 41, 42sylancl 693 . . . . . . . . . . 11 ((suc 𝑥𝐵𝐵𝑥) → suc 𝑥𝑥)
4443expcom 451 . . . . . . . . . 10 (𝐵𝑥 → (suc 𝑥𝐵 → suc 𝑥𝑥))
45 peano2b 7029 . . . . . . . . . . . . 13 (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
46 nnord 7021 . . . . . . . . . . . . 13 (suc 𝑥 ∈ ω → Ord suc 𝑥)
4745, 46sylbi 207 . . . . . . . . . . . 12 (𝑥 ∈ ω → Ord suc 𝑥)
4820sucid 5766 . . . . . . . . . . . 12 𝑥 ∈ suc 𝑥
49 nordeq 5704 . . . . . . . . . . . 12 ((Ord suc 𝑥𝑥 ∈ suc 𝑥) → suc 𝑥𝑥)
5047, 48, 49sylancl 693 . . . . . . . . . . 11 (𝑥 ∈ ω → suc 𝑥𝑥)
51 nneneq 8088 . . . . . . . . . . . . . 14 ((suc 𝑥 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑥𝑥 ↔ suc 𝑥 = 𝑥))
5245, 51sylanb 489 . . . . . . . . . . . . 13 ((𝑥 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑥𝑥 ↔ suc 𝑥 = 𝑥))
5352anidms 676 . . . . . . . . . . . 12 (𝑥 ∈ ω → (suc 𝑥𝑥 ↔ suc 𝑥 = 𝑥))
5453necon3bbid 2833 . . . . . . . . . . 11 (𝑥 ∈ ω → (¬ suc 𝑥𝑥 ↔ suc 𝑥𝑥))
5550, 54mpbird 247 . . . . . . . . . 10 (𝑥 ∈ ω → ¬ suc 𝑥𝑥)
5644, 55nsyli 155 . . . . . . . . 9 (𝐵𝑥 → (𝑥 ∈ ω → ¬ suc 𝑥𝐵))
5737, 56syli 39 . . . . . . . 8 (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → ¬ suc 𝑥𝐵))
5857com12 32 . . . . . . 7 (𝑥 ∈ ω → (𝐵 ⊊ suc 𝑥 → ¬ suc 𝑥𝐵))
59 psseq2 3678 . . . . . . . 8 (𝐴 = suc 𝑥 → (𝐵𝐴𝐵 ⊊ suc 𝑥))
60 breq1 4621 . . . . . . . . 9 (𝐴 = suc 𝑥 → (𝐴𝐵 ↔ suc 𝑥𝐵))
6160notbid 308 . . . . . . . 8 (𝐴 = suc 𝑥 → (¬ 𝐴𝐵 ↔ ¬ suc 𝑥𝐵))
6259, 61imbi12d 334 . . . . . . 7 (𝐴 = suc 𝑥 → ((𝐵𝐴 → ¬ 𝐴𝐵) ↔ (𝐵 ⊊ suc 𝑥 → ¬ suc 𝑥𝐵)))
6358, 62syl5ibrcom 237 . . . . . 6 (𝑥 ∈ ω → (𝐴 = suc 𝑥 → (𝐵𝐴 → ¬ 𝐴𝐵)))
6463rexlimiv 3025 . . . . 5 (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → (𝐵𝐴 → ¬ 𝐴𝐵))
6510, 64syl 17 . . . 4 ((𝐴 ∈ ω ∧ 𝐵𝐴) → (𝐵𝐴 → ¬ 𝐴𝐵))
6665ex 450 . . 3 (𝐴 ∈ ω → (𝐵𝐴 → (𝐵𝐴 → ¬ 𝐴𝐵)))
6766pm2.43d 53 . 2 (𝐴 ∈ ω → (𝐵𝐴 → ¬ 𝐴𝐵))
6867imp 445 1 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ¬ 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1992  wne 2796  wrex 2913  Vcvv 3191  cdif 3557  cin 3559  wss 3560  wpss 3561  c0 3896  {csn 4153   class class class wbr 4618  Ord word 5684  suc csuc 5687  ωcom 7013  cen 7897  cdom 7898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-er 7688  df-en 7901  df-dom 7902
This theorem is referenced by:  php2  8090  php3  8091
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