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Theorem phpm 6864
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 6851 through phplem4 6854, nneneq 6856, 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 3257 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐴)
3 ne0i 3429 . . . . . . . . 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 6862 . . . . . . . . . 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 2740 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
1312sucex 4498 . . . . . . . . . . . . . . 15 suc 𝑦 ∈ V
14 difss 3261 . . . . . . . . . . . . . . 15 (suc 𝑦 ∖ {𝑥}) ⊆ suc 𝑦
1513, 14ssexi 4141 . . . . . . . . . . . . . 14 (suc 𝑦 ∖ {𝑥}) ∈ V
16 eldifn 3258 . . . . . . . . . . . . . . . 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 3193 . . . . . . . . . . . . . . 15 (((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵 ⊆ suc 𝑦)
22 ssdif 3270 . . . . . . . . . . . . . . . 16 (𝐵 ⊆ suc 𝑦 → (𝐵 ∖ {𝑥}) ⊆ (suc 𝑦 ∖ {𝑥}))
23 disjsn 3654 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥𝐵)
24 disj3 3475 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∩ {𝑥}) = ∅ ↔ 𝐵 = (𝐵 ∖ {𝑥}))
2523, 24bitr3i 186 . . . . . . . . . . . . . . . . 17 𝑥𝐵𝐵 = (𝐵 ∖ {𝑥}))
26 sseq1 3178 . . . . . . . . . . . . . . . . 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 6777 . . . . . . . . . . . . . 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 6855 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ω ∧ 𝑥 ∈ suc 𝑦) → 𝑦 ≈ (suc 𝑦 ∖ {𝑥}))
3635ensymd 6782 . . . . . . . . . . . . . 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 6790 . . . . . . . . . . . . 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 6789 . . . . . . . . . . 11 ((𝐴𝐵𝐵𝑦) → 𝐴𝑦)
4211, 40, 41syl2anc 411 . . . . . . . . . 10 ((((((𝐴 ∈ ω ∧ 𝐵𝐴) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴𝐵) → 𝐴𝑦)
4310, 42eqbrtrrd 4027 . . . . . . . . 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 4603 . . . . . 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 2737  cdif 3126  cin 3128  wss 3129  c0 3422  {csn 3592   class class class wbr 4003  suc csuc 4365  ωcom 4589  cen 6737  cdom 6738
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 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-er 6534  df-en 6740  df-dom 6741
This theorem is referenced by:  phpelm  6865
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