Theorem phpm 6921

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 6908 through phplem4 6911, nneneq 6913, 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 3281	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐴)
3		ne0i 3453	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
4	2, 3	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝐴 ≠ ∅)
5	4	neneqd 2385	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝐴 = ∅)
6	5	ad2antlr 489	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝐴 = ∅) → ¬ 𝐴 = ∅)
7	1, 6	pm2.21dd 621	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝐴 = ∅) → ¬ 𝐴 ≈ 𝐵)
8		php5dom 6919	. . . . . . . . . 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 2763	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
13	12	sucex 4531	. . . . . . . . . . . . . . 15 ⊢ suc 𝑦 ∈ V
14		difss 3285	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑦 ∖ {𝑥}) ⊆ suc 𝑦
15	13, 14	ssexi 4167	. . . . . . . . . . . . . 14 ⊢ (suc 𝑦 ∖ {𝑥}) ∈ V
16		eldifn 3282	. . . . . . . . . . . . . . . 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 3217	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵 ⊆ suc 𝑦)
22		ssdif 3294	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ⊆ suc 𝑦 → (𝐵 ∖ {𝑥}) ⊆ (suc 𝑦 ∖ {𝑥}))
23		disjsn 3680	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝐵)
24		disj3 3499	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∩ {𝑥}) = ∅ ↔ 𝐵 = (𝐵 ∖ {𝑥}))
25	23, 24	bitr3i 186	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ 𝐵 = (𝐵 ∖ {𝑥}))
26		sseq1 3202	. . . . . . . . . . . . . . . . 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 6832	. . . . . . . . . . . . . 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 2272	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝑥 ∈ suc 𝑦)
35		phplem3g 6912	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ω ∧ 𝑥 ∈ suc 𝑦) → 𝑦 ≈ (suc 𝑦 ∖ {𝑥}))
36	35	ensymd 6837	. . . . . . . . . . . . . 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 6845	. . . . . . . . . . . . 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 6844	. . . . . . . . . . 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 4053	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴 ≈ 𝐵) → suc 𝑦 ≼ 𝑦)
44	9, 43	mtand 666	. . . . . . . 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 2611	. . . . . 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 4636	. . . . . 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 799	. . . 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 1830	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝐴 ≈ 𝐵))
53	52	3impia 1202	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) → ¬ 𝐴 ≈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∧ w3a 980   = wceq 1364  ∃wex 1503   ∈ wcel 2164   ≠ wne 2364  ∃wrex 2473  Vcvv 2760   ∖ cdif 3150   ∩ cin 3152   ⊆ wss 3153  ∅c0 3446  {csn 3618   class class class wbr 4029  suc csuc 4396  ωcom 4622   ≈ cen 6792   ≼ cdom 6793

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-er 6587  df-en 6795  df-dom 6796

This theorem is referenced by:  phpelm  6922