ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  phpm GIF version

Theorem phpm 6865
Description: Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. By "proper subset" here we mean that there is an element which is in the natural number and not in the subset, or in symbols 𝑥𝑥 ∈ (𝐴𝐵) (which is stronger than not being equal in the absence of excluded middle). 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 6852 through phplem4 6855, nneneq 6857, and this final piece of the proof. (Contributed by NM, 29-May-1998.)
Assertion
Ref Expression
phpm ((𝐴 ∈ ω ∧ 𝐵𝐴 ∧ ∃𝑥 𝑥 ∈ (𝐴𝐵)) → ¬ 𝐴𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem phpm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpr 110 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝐴 = ∅) → 𝐴 = ∅)
2 eldifi 3258 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐴)
3 ne0i 3430 . . . . . . . . 9 (𝑥𝐴𝐴 ≠ ∅)
42, 3syl 14 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) → 𝐴 ≠ ∅)
54neneqd 2368 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → ¬ 𝐴 = ∅)
65ad2antlr 489 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝐴 = ∅) → ¬ 𝐴 = ∅)
71, 6pm2.21dd 620 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝐴 = ∅) → ¬ 𝐴𝐵)
8 php5dom 6863 . . . . . . . . . 10 (𝑦 ∈ ω → ¬ suc 𝑦𝑦)
98ad2antlr 489 . . . . . . . . 9 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → ¬ suc 𝑦𝑦)
10 simplr 528 . . . . . . . . . 10 ((((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴𝐵) → 𝐴 = suc 𝑦)
11 simpr 110 . . . . . . . . . . 11 ((((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴𝐵) → 𝐴𝐵)
12 vex 2741 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
1312sucex 4499 . . . . . . . . . . . . . . 15 suc 𝑦 ∈ V
14 difss 3262 . . . . . . . . . . . . . . 15 (suc 𝑦 ∖ {𝑥}) ⊆ suc 𝑦
1513, 14ssexi 4142 . . . . . . . . . . . . . 14 (suc 𝑦 ∖ {𝑥}) ∈ V
16 eldifn 3259 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥𝐵)
1716ad3antlr 493 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → ¬ 𝑥𝐵)
18 simpllr 534 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) → 𝐵𝐴)
1918adantr 276 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵𝐴)
20 simpr 110 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐴 = suc 𝑦)
2119, 20sseqtrd 3194 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵 ⊆ suc 𝑦)
22 ssdif 3271 . . . . . . . . . . . . . . . 16 (𝐵 ⊆ suc 𝑦 → (𝐵 ∖ {𝑥}) ⊆ (suc 𝑦 ∖ {𝑥}))
23 disjsn 3655 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥𝐵)
24 disj3 3476 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∩ {𝑥}) = ∅ ↔ 𝐵 = (𝐵 ∖ {𝑥}))
2523, 24bitr3i 186 . . . . . . . . . . . . . . . . 17 𝑥𝐵𝐵 = (𝐵 ∖ {𝑥}))
26 sseq1 3179 . . . . . . . . . . . . . . . . 17 (𝐵 = (𝐵 ∖ {𝑥}) → (𝐵 ⊆ (suc 𝑦 ∖ {𝑥}) ↔ (𝐵 ∖ {𝑥}) ⊆ (suc 𝑦 ∖ {𝑥})))
2725, 26sylbi 121 . . . . . . . . . . . . . . . 16 𝑥𝐵 → (𝐵 ⊆ (suc 𝑦 ∖ {𝑥}) ↔ (𝐵 ∖ {𝑥}) ⊆ (suc 𝑦 ∖ {𝑥})))
2822, 27imbitrrid 156 . . . . . . . . . . . . . . 15 𝑥𝐵 → (𝐵 ⊆ suc 𝑦𝐵 ⊆ (suc 𝑦 ∖ {𝑥})))
2917, 21, 28sylc 62 . . . . . . . . . . . . . 14 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵 ⊆ (suc 𝑦 ∖ {𝑥}))
30 ssdomg 6778 . . . . . . . . . . . . . 14 ((suc 𝑦 ∖ {𝑥}) ∈ V → (𝐵 ⊆ (suc 𝑦 ∖ {𝑥}) → 𝐵 ≼ (suc 𝑦 ∖ {𝑥})))
3115, 29, 30mpsyl 65 . . . . . . . . . . . . 13 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵 ≼ (suc 𝑦 ∖ {𝑥}))
32 simplr 528 . . . . . . . . . . . . . 14 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝑦 ∈ ω)
332ad3antlr 493 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝑥𝐴)
3433, 20eleqtrd 2256 . . . . . . . . . . . . . 14 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝑥 ∈ suc 𝑦)
35 phplem3g 6856 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ω ∧ 𝑥 ∈ suc 𝑦) → 𝑦 ≈ (suc 𝑦 ∖ {𝑥}))
3635ensymd 6783 . . . . . . . . . . . . . 14 ((𝑦 ∈ ω ∧ 𝑥 ∈ suc 𝑦) → (suc 𝑦 ∖ {𝑥}) ≈ 𝑦)
3732, 34, 36syl2anc 411 . . . . . . . . . . . . 13 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → (suc 𝑦 ∖ {𝑥}) ≈ 𝑦)
38 domentr 6791 . . . . . . . . . . . . 13 ((𝐵 ≼ (suc 𝑦 ∖ {𝑥}) ∧ (suc 𝑦 ∖ {𝑥}) ≈ 𝑦) → 𝐵𝑦)
3931, 37, 38syl2anc 411 . . . . . . . . . . . 12 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵𝑦)
4039adantr 276 . . . . . . . . . . 11 ((((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴𝐵) → 𝐵𝑦)
41 endomtr 6790 . . . . . . . . . . 11 ((𝐴𝐵𝐵𝑦) → 𝐴𝑦)
4211, 40, 41syl2anc 411 . . . . . . . . . 10 ((((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴𝐵) → 𝐴𝑦)
4310, 42eqbrtrrd 4028 . . . . . . . . 9 ((((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴𝐵) → suc 𝑦𝑦)
449, 43mtand 665 . . . . . . . 8 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → ¬ 𝐴𝐵)
4544ex 115 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) → (𝐴 = suc 𝑦 → ¬ 𝐴𝐵))
4645rexlimdva 2594 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) → (∃𝑦 ∈ ω 𝐴 = suc 𝑦 → ¬ 𝐴𝐵))
4746imp 124 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ∃𝑦 ∈ ω 𝐴 = suc 𝑦) → ¬ 𝐴𝐵)
48 nn0suc 4604 . . . . . 6 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑦 ∈ ω 𝐴 = suc 𝑦))
4948ad2antrr 488 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) → (𝐴 = ∅ ∨ ∃𝑦 ∈ ω 𝐴 = suc 𝑦))
507, 47, 49mpjaodan 798 . . . 4 (((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) → ¬ 𝐴𝐵)
5150ex 115 . . 3 ((𝐴 ∈ ω ∧ 𝐵𝐴) → (𝑥 ∈ (𝐴𝐵) → ¬ 𝐴𝐵))
5251exlimdv 1819 . 2 ((𝐴 ∈ ω ∧ 𝐵𝐴) → (∃𝑥 𝑥 ∈ (𝐴𝐵) → ¬ 𝐴𝐵))
53523impia 1200 1 ((𝐴 ∈ ω ∧ 𝐵𝐴 ∧ ∃𝑥 𝑥 ∈ (𝐴𝐵)) → ¬ 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  w3a 978   = wceq 1353  wex 1492  wcel 2148  wne 2347  wrex 2456  Vcvv 2738  cdif 3127  cin 3129  wss 3130  c0 3423  {csn 3593   class class class wbr 4004  suc csuc 4366  ωcom 4590  cen 6738  cdom 6739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-er 6535  df-en 6741  df-dom 6742
This theorem is referenced by:  phpelm  6866
  Copyright terms: Public domain W3C validator