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Theorem php 8301
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 8296 through phplem4 8299, nneneq 8300, 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 4117 . . . . . . . 8 ∅ ⊆ 𝐵
2 sspsstr 3863 . . . . . . . 8 ((∅ ⊆ 𝐵𝐵𝐴) → ∅ ⊊ 𝐴)
31, 2mpan 664 . . . . . . 7 (𝐵𝐴 → ∅ ⊊ 𝐴)
4 0pss 4158 . . . . . . . 8 (∅ ⊊ 𝐴𝐴 ≠ ∅)
5 df-ne 2944 . . . . . . . 8 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
64, 5bitri 264 . . . . . . 7 (∅ ⊊ 𝐴 ↔ ¬ 𝐴 = ∅)
73, 6sylib 208 . . . . . 6 (𝐵𝐴 → ¬ 𝐴 = ∅)
8 nn0suc 7238 . . . . . . 7 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
98orcanai 971 . . . . . 6 ((𝐴 ∈ ω ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
107, 9sylan2 574 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
11 pssnel 4182 . . . . . . . . . 10 (𝐵 ⊊ suc 𝑥 → ∃𝑦(𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦𝐵))
12 pssss 3853 . . . . . . . . . . . . . . . . 17 (𝐵 ⊊ suc 𝑥𝐵 ⊆ suc 𝑥)
13 ssdif 3897 . . . . . . . . . . . . . . . . . 18 (𝐵 ⊆ suc 𝑥 → (𝐵 ∖ {𝑦}) ⊆ (suc 𝑥 ∖ {𝑦}))
14 disjsn 4384 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝐵)
15 disj3 4165 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∩ {𝑦}) = ∅ ↔ 𝐵 = (𝐵 ∖ {𝑦}))
1614, 15bitr3i 266 . . . . . . . . . . . . . . . . . . 19 𝑦𝐵𝐵 = (𝐵 ∖ {𝑦}))
17 sseq1 3776 . . . . . . . . . . . . . . . . . . 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 3354 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ V
2120sucex 7159 . . . . . . . . . . . . . . . . . . 19 suc 𝑥 ∈ V
22 difss 3889 . . . . . . . . . . . . . . . . . . 19 (suc 𝑥 ∖ {𝑦}) ⊆ suc 𝑥
2321, 22ssexi 4938 . . . . . . . . . . . . . . . . . 18 (suc 𝑥 ∖ {𝑦}) ∈ V
24 ssdomg 8156 . . . . . . . . . . . . . . . . . 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 393 . . . . . . . . . . . . . . 15 ((¬ 𝑦𝐵𝐵 ⊊ suc 𝑥) → 𝐵 ≼ (suc 𝑥 ∖ {𝑦}))
28 vex 3354 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
2920, 28phplem3 8298 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥) → 𝑥 ≈ (suc 𝑥 ∖ {𝑦}))
3029ensymd 8161 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥) → (suc 𝑥 ∖ {𝑦}) ≈ 𝑥)
31 domentr 8169 . . . . . . . . . . . . . . 15 ((𝐵 ≼ (suc 𝑥 ∖ {𝑦}) ∧ (suc 𝑥 ∖ {𝑦}) ≈ 𝑥) → 𝐵𝑥)
3227, 30, 31syl2an 577 . . . . . . . . . . . . . 14 (((¬ 𝑦𝐵𝐵 ⊊ suc 𝑥) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥)) → 𝐵𝑥)
3332exp43 423 . . . . . . . . . . . . 13 𝑦𝐵 → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → (𝑦 ∈ suc 𝑥𝐵𝑥))))
3433com4r 94 . . . . . . . . . . . 12 (𝑦 ∈ suc 𝑥 → (¬ 𝑦𝐵 → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵𝑥))))
3534imp 393 . . . . . . . . . . 11 ((𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦𝐵) → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵𝑥)))
3635exlimiv 2010 . . . . . . . . . 10 (∃𝑦(𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦𝐵) → (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵𝑥)))
3711, 36mpcom 38 . . . . . . . . 9 (𝐵 ⊊ suc 𝑥 → (𝑥 ∈ ω → 𝐵𝑥))
38 endomtr 8168 . . . . . . . . . . . 12 ((suc 𝑥𝐵𝐵𝑥) → suc 𝑥𝑥)
39 sssucid 5946 . . . . . . . . . . . . 13 𝑥 ⊆ suc 𝑥
40 ssdomg 8156 . . . . . . . . . . . . 13 (suc 𝑥 ∈ V → (𝑥 ⊆ suc 𝑥𝑥 ≼ suc 𝑥))
4121, 39, 40mp2 9 . . . . . . . . . . . 12 𝑥 ≼ suc 𝑥
42 sbth 8237 . . . . . . . . . . . 12 ((suc 𝑥𝑥𝑥 ≼ suc 𝑥) → suc 𝑥𝑥)
4338, 41, 42sylancl 568 . . . . . . . . . . 11 ((suc 𝑥𝐵𝐵𝑥) → suc 𝑥𝑥)
4443expcom 398 . . . . . . . . . 10 (𝐵𝑥 → (suc 𝑥𝐵 → suc 𝑥𝑥))
45 peano2b 7229 . . . . . . . . . . . . 13 (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
46 nnord 7221 . . . . . . . . . . . . 13 (suc 𝑥 ∈ ω → Ord suc 𝑥)
4745, 46sylbi 207 . . . . . . . . . . . 12 (𝑥 ∈ ω → Ord suc 𝑥)
4820sucid 5948 . . . . . . . . . . . 12 𝑥 ∈ suc 𝑥
49 nordeq 5886 . . . . . . . . . . . 12 ((Ord suc 𝑥𝑥 ∈ suc 𝑥) → suc 𝑥𝑥)
5047, 48, 49sylancl 568 . . . . . . . . . . 11 (𝑥 ∈ ω → suc 𝑥𝑥)
51 nneneq 8300 . . . . . . . . . . . . . 14 ((suc 𝑥 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑥𝑥 ↔ suc 𝑥 = 𝑥))
5245, 51sylanb 564 . . . . . . . . . . . . 13 ((𝑥 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑥𝑥 ↔ suc 𝑥 = 𝑥))
5352anidms 550 . . . . . . . . . . . 12 (𝑥 ∈ ω → (suc 𝑥𝑥 ↔ suc 𝑥 = 𝑥))
5453necon3bbid 2980 . . . . . . . . . . 11 (𝑥 ∈ ω → (¬ suc 𝑥𝑥 ↔ suc 𝑥𝑥))
5550, 54mpbird 247 . . . . . . . . . 10 (𝑥 ∈ ω → ¬ suc 𝑥𝑥)
5644, 55nsyli 156 . . . . . . . . 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 3846 . . . . . . . 8 (𝐴 = suc 𝑥 → (𝐵𝐴𝐵 ⊊ suc 𝑥))
60 breq1 4790 . . . . . . . . 9 (𝐴 = suc 𝑥 → (𝐴𝐵 ↔ suc 𝑥𝐵))
6160notbid 307 . . . . . . . 8 (𝐴 = suc 𝑥 → (¬ 𝐴𝐵 ↔ ¬ suc 𝑥𝐵))
6259, 61imbi12d 333 . . . . . . 7 (𝐴 = suc 𝑥 → ((𝐵𝐴 → ¬ 𝐴𝐵) ↔ (𝐵 ⊊ suc 𝑥 → ¬ suc 𝑥𝐵)))
6358, 62syl5ibrcom 237 . . . . . 6 (𝑥 ∈ ω → (𝐴 = suc 𝑥 → (𝐵𝐴 → ¬ 𝐴𝐵)))
6463rexlimiv 3175 . . . . 5 (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → (𝐵𝐴 → ¬ 𝐴𝐵))
6510, 64syl 17 . . . 4 ((𝐴 ∈ ω ∧ 𝐵𝐴) → (𝐵𝐴 → ¬ 𝐴𝐵))
6665ex 397 . . 3 (𝐴 ∈ ω → (𝐵𝐴 → (𝐵𝐴 → ¬ 𝐴𝐵)))
6766pm2.43d 53 . 2 (𝐴 ∈ ω → (𝐵𝐴 → ¬ 𝐴𝐵))
6867imp 393 1 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ¬ 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  wne 2943  wrex 3062  Vcvv 3351  cdif 3721  cin 3723  wss 3724  wpss 3725  c0 4064  {csn 4317   class class class wbr 4787  Ord word 5866  suc csuc 5869  ωcom 7213  cen 8107  cdom 8108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-om 7214  df-er 7897  df-en 8111  df-dom 8112
This theorem is referenced by:  php2  8302  php3  8303
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