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Theorem phpm 6511
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 6498 through phplem4 6501, nneneq 6503, 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 108 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝐴 = ∅) → 𝐴 = ∅)
2 eldifi 3106 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐴)
3 ne0i 3275 . . . . . . . . 9 (𝑥𝐴𝐴 ≠ ∅)
42, 3syl 14 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) → 𝐴 ≠ ∅)
54neneqd 2270 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → ¬ 𝐴 = ∅)
65ad2antlr 473 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝐴 = ∅) → ¬ 𝐴 = ∅)
71, 6pm2.21dd 583 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝐴 = ∅) → ¬ 𝐴𝐵)
8 php5dom 6509 . . . . . . . . . 10 (𝑦 ∈ ω → ¬ suc 𝑦𝑦)
98ad2antlr 473 . . . . . . . . 9 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → ¬ suc 𝑦𝑦)
10 simplr 497 . . . . . . . . . 10 ((((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴𝐵) → 𝐴 = suc 𝑦)
11 simpr 108 . . . . . . . . . . 11 ((((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴𝐵) → 𝐴𝐵)
12 vex 2615 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
1312sucex 4279 . . . . . . . . . . . . . . 15 suc 𝑦 ∈ V
14 difss 3110 . . . . . . . . . . . . . . 15 (suc 𝑦 ∖ {𝑥}) ⊆ suc 𝑦
1513, 14ssexi 3942 . . . . . . . . . . . . . 14 (suc 𝑦 ∖ {𝑥}) ∈ V
16 eldifn 3107 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥𝐵)
1716ad3antlr 477 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → ¬ 𝑥𝐵)
18 simpllr 501 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) → 𝐵𝐴)
1918adantr 270 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵𝐴)
20 simpr 108 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐴 = suc 𝑦)
2119, 20sseqtrd 3046 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵 ⊆ suc 𝑦)
22 ssdif 3119 . . . . . . . . . . . . . . . 16 (𝐵 ⊆ suc 𝑦 → (𝐵 ∖ {𝑥}) ⊆ (suc 𝑦 ∖ {𝑥}))
23 disjsn 3478 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥𝐵)
24 disj3 3317 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∩ {𝑥}) = ∅ ↔ 𝐵 = (𝐵 ∖ {𝑥}))
2523, 24bitr3i 184 . . . . . . . . . . . . . . . . 17 𝑥𝐵𝐵 = (𝐵 ∖ {𝑥}))
26 sseq1 3031 . . . . . . . . . . . . . . . . 17 (𝐵 = (𝐵 ∖ {𝑥}) → (𝐵 ⊆ (suc 𝑦 ∖ {𝑥}) ↔ (𝐵 ∖ {𝑥}) ⊆ (suc 𝑦 ∖ {𝑥})))
2725, 26sylbi 119 . . . . . . . . . . . . . . . 16 𝑥𝐵 → (𝐵 ⊆ (suc 𝑦 ∖ {𝑥}) ↔ (𝐵 ∖ {𝑥}) ⊆ (suc 𝑦 ∖ {𝑥})))
2822, 27syl5ibr 154 . . . . . . . . . . . . . . 15 𝑥𝐵 → (𝐵 ⊆ suc 𝑦𝐵 ⊆ (suc 𝑦 ∖ {𝑥})))
2917, 21, 28sylc 61 . . . . . . . . . . . . . 14 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵 ⊆ (suc 𝑦 ∖ {𝑥}))
30 ssdomg 6425 . . . . . . . . . . . . . 14 ((suc 𝑦 ∖ {𝑥}) ∈ V → (𝐵 ⊆ (suc 𝑦 ∖ {𝑥}) → 𝐵 ≼ (suc 𝑦 ∖ {𝑥})))
3115, 29, 30mpsyl 64 . . . . . . . . . . . . 13 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵 ≼ (suc 𝑦 ∖ {𝑥}))
32 simplr 497 . . . . . . . . . . . . . 14 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝑦 ∈ ω)
332ad3antlr 477 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝑥𝐴)
3433, 20eleqtrd 2161 . . . . . . . . . . . . . 14 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝑥 ∈ suc 𝑦)
35 phplem3g 6502 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ω ∧ 𝑥 ∈ suc 𝑦) → 𝑦 ≈ (suc 𝑦 ∖ {𝑥}))
3635ensymd 6430 . . . . . . . . . . . . . 14 ((𝑦 ∈ ω ∧ 𝑥 ∈ suc 𝑦) → (suc 𝑦 ∖ {𝑥}) ≈ 𝑦)
3732, 34, 36syl2anc 403 . . . . . . . . . . . . 13 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → (suc 𝑦 ∖ {𝑥}) ≈ 𝑦)
38 domentr 6438 . . . . . . . . . . . . 13 ((𝐵 ≼ (suc 𝑦 ∖ {𝑥}) ∧ (suc 𝑦 ∖ {𝑥}) ≈ 𝑦) → 𝐵𝑦)
3931, 37, 38syl2anc 403 . . . . . . . . . . . 12 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵𝑦)
4039adantr 270 . . . . . . . . . . 11 ((((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴𝐵) → 𝐵𝑦)
41 endomtr 6437 . . . . . . . . . . 11 ((𝐴𝐵𝐵𝑦) → 𝐴𝑦)
4211, 40, 41syl2anc 403 . . . . . . . . . 10 ((((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴𝐵) → 𝐴𝑦)
4310, 42eqbrtrrd 3833 . . . . . . . . 9 ((((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴𝐵) → suc 𝑦𝑦)
449, 43mtand 624 . . . . . . . 8 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → ¬ 𝐴𝐵)
4544ex 113 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) → (𝐴 = suc 𝑦 → ¬ 𝐴𝐵))
4645rexlimdva 2483 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) → (∃𝑦 ∈ ω 𝐴 = suc 𝑦 → ¬ 𝐴𝐵))
4746imp 122 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ∃𝑦 ∈ ω 𝐴 = suc 𝑦) → ¬ 𝐴𝐵)
48 nn0suc 4382 . . . . . 6 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑦 ∈ ω 𝐴 = suc 𝑦))
4948ad2antrr 472 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) → (𝐴 = ∅ ∨ ∃𝑦 ∈ ω 𝐴 = suc 𝑦))
507, 47, 49mpjaodan 745 . . . 4 (((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) → ¬ 𝐴𝐵)
5150ex 113 . . 3 ((𝐴 ∈ ω ∧ 𝐵𝐴) → (𝑥 ∈ (𝐴𝐵) → ¬ 𝐴𝐵))
5251exlimdv 1742 . 2 ((𝐴 ∈ ω ∧ 𝐵𝐴) → (∃𝑥 𝑥 ∈ (𝐴𝐵) → ¬ 𝐴𝐵))
53523impia 1136 1 ((𝐴 ∈ ω ∧ 𝐵𝐴 ∧ ∃𝑥 𝑥 ∈ (𝐴𝐵)) → ¬ 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  w3a 920   = wceq 1285  wex 1422  wcel 1434  wne 2249  wrex 2354  Vcvv 2612  cdif 2981  cin 2983  wss 2984  c0 3269  {csn 3422   class class class wbr 3811  suc csuc 4156  ωcom 4368  cen 6385  cdom 6386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-br 3812  df-opab 3866  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-er 6222  df-en 6388  df-dom 6389
This theorem is referenced by:  phpelm  6512
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