Theorem phpm 7023

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 ∈ (𝐴 ∖ 𝐵) (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 7009 through phplem4 7012, nneneq 7014, and this final piece of the proof. (Contributed by NM, 29-May-1998.)

Assertion

Ref	Expression
phpm	⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) → ¬ 𝐴 ≈ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem phpm

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝐴 = ∅) → 𝐴 = ∅)
2		eldifi 3326	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐴)
3		ne0i 3498	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
4	2, 3	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝐴 ≠ ∅)
5	4	neneqd 2421	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝐴 = ∅)
6	5	ad2antlr 489	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝐴 = ∅) → ¬ 𝐴 = ∅)
7	1, 6	pm2.21dd 623	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝐴 = ∅) → ¬ 𝐴 ≈ 𝐵)
8		php5dom 7020	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → ¬ suc 𝑦 ≼ 𝑦)
9	8	ad2antlr 489	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → ¬ suc 𝑦 ≼ 𝑦)
10		simplr 528	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴 ≈ 𝐵) → 𝐴 = suc 𝑦)
11		simpr 110	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴 ≈ 𝐵) → 𝐴 ≈ 𝐵)
12		vex 2802	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
13	12	sucex 4590	. . . . . . . . . . . . . . 15 ⊢ suc 𝑦 ∈ V
14		difss 3330	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑦 ∖ {𝑥}) ⊆ suc 𝑦
15	13, 14	ssexi 4221	. . . . . . . . . . . . . 14 ⊢ (suc 𝑦 ∖ {𝑥}) ∈ V
16		eldifn 3327	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵)
17	16	ad3antlr 493	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → ¬ 𝑥 ∈ 𝐵)
18		simpllr 534	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) → 𝐵 ⊆ 𝐴)
19	18	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵 ⊆ 𝐴)
20		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐴 = suc 𝑦)
21	19, 20	sseqtrd 3262	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵 ⊆ suc 𝑦)
22		ssdif 3339	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ⊆ suc 𝑦 → (𝐵 ∖ {𝑥}) ⊆ (suc 𝑦 ∖ {𝑥}))
23		disjsn 3728	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝐵)
24		disj3 3544	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∩ {𝑥}) = ∅ ↔ 𝐵 = (𝐵 ∖ {𝑥}))
25	23, 24	bitr3i 186	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ 𝐵 = (𝐵 ∖ {𝑥}))
26		sseq1 3247	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = (𝐵 ∖ {𝑥}) → (𝐵 ⊆ (suc 𝑦 ∖ {𝑥}) ↔ (𝐵 ∖ {𝑥}) ⊆ (suc 𝑦 ∖ {𝑥})))
27	25, 26	sylbi 121	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 ∈ 𝐵 → (𝐵 ⊆ (suc 𝑦 ∖ {𝑥}) ↔ (𝐵 ∖ {𝑥}) ⊆ (suc 𝑦 ∖ {𝑥})))
28	22, 27	imbitrrid 156	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥 ∈ 𝐵 → (𝐵 ⊆ suc 𝑦 → 𝐵 ⊆ (suc 𝑦 ∖ {𝑥})))
29	17, 21, 28	sylc 62	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵 ⊆ (suc 𝑦 ∖ {𝑥}))
30		ssdomg 6928	. . . . . . . . . . . . . 14 ⊢ ((suc 𝑦 ∖ {𝑥}) ∈ V → (𝐵 ⊆ (suc 𝑦 ∖ {𝑥}) → 𝐵 ≼ (suc 𝑦 ∖ {𝑥})))
31	15, 29, 30	mpsyl 65	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵 ≼ (suc 𝑦 ∖ {𝑥}))
32		simplr 528	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝑦 ∈ ω)
33	2	ad3antlr 493	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝑥 ∈ 𝐴)
34	33, 20	eleqtrd 2308	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝑥 ∈ suc 𝑦)
35		phplem3g 7013	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ω ∧ 𝑥 ∈ suc 𝑦) → 𝑦 ≈ (suc 𝑦 ∖ {𝑥}))
36	35	ensymd 6933	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ω ∧ 𝑥 ∈ suc 𝑦) → (suc 𝑦 ∖ {𝑥}) ≈ 𝑦)
37	32, 34, 36	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → (suc 𝑦 ∖ {𝑥}) ≈ 𝑦)
38		domentr 6941	. . . . . . . . . . . . 13 ⊢ ((𝐵 ≼ (suc 𝑦 ∖ {𝑥}) ∧ (suc 𝑦 ∖ {𝑥}) ≈ 𝑦) → 𝐵 ≼ 𝑦)
39	31, 37, 38	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵 ≼ 𝑦)
40	39	adantr 276	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴 ≈ 𝐵) → 𝐵 ≼ 𝑦)
41		endomtr 6940	. . . . . . . . . . 11 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≼ 𝑦) → 𝐴 ≼ 𝑦)
42	11, 40, 41	syl2anc 411	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴 ≈ 𝐵) → 𝐴 ≼ 𝑦)
43	10, 42	eqbrtrrd 4106	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴 ≈ 𝐵) → suc 𝑦 ≼ 𝑦)
44	9, 43	mtand 669	. . . . . . . 8 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → ¬ 𝐴 ≈ 𝐵)
45	44	ex 115	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) → (𝐴 = suc 𝑦 → ¬ 𝐴 ≈ 𝐵))
46	45	rexlimdva 2648	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) → (∃𝑦 ∈ ω 𝐴 = suc 𝑦 → ¬ 𝐴 ≈ 𝐵))
47	46	imp 124	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ ∃𝑦 ∈ ω 𝐴 = suc 𝑦) → ¬ 𝐴 ≈ 𝐵)
48		nn0suc 4695	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑦 ∈ ω 𝐴 = suc 𝑦))
49	48	ad2antrr 488	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) → (𝐴 = ∅ ∨ ∃𝑦 ∈ ω 𝐴 = suc 𝑦))
50	7, 47, 49	mpjaodan 803	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) → ¬ 𝐴 ≈ 𝐵)
51	50	ex 115	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝐴 ≈ 𝐵))
52	51	exlimdv 1865	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝐴 ≈ 𝐵))
53	52	3impia 1224	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) → ¬ 𝐴 ≈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   ∧ w3a 1002   = wceq 1395  ∃wex 1538   ∈ wcel 2200   ≠ wne 2400  ∃wrex 2509  Vcvv 2799   ∖ cdif 3194   ∩ cin 3196   ⊆ wss 3197  ∅c0 3491  {csn 3666   class class class wbr 4082  suc csuc 4455  ωcom 4681   ≈ cen 6883   ≼ cdom 6884

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-er 6678  df-en 6886  df-dom 6887

This theorem is referenced by:  phpelm  7024