Theorem phpm 6882

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 6869 through phplem4 6872, nneneq 6874, 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 3271	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐴)
3		ne0i 3443	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
4	2, 3	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝐴 ≠ ∅)
5	4	neneqd 2380	. . . . . . 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 6880	. . . . . . . . . 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 2754	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
13	12	sucex 4512	. . . . . . . . . . . . . . 15 ⊢ suc 𝑦 ∈ V
14		difss 3275	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑦 ∖ {𝑥}) ⊆ suc 𝑦
15	13, 14	ssexi 4155	. . . . . . . . . . . . . 14 ⊢ (suc 𝑦 ∖ {𝑥}) ∈ V
16		eldifn 3272	. . . . . . . . . . . . . . . 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 3207	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵 ⊆ suc 𝑦)
22		ssdif 3284	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ⊆ suc 𝑦 → (𝐵 ∖ {𝑥}) ⊆ (suc 𝑦 ∖ {𝑥}))
23		disjsn 3668	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝐵)
24		disj3 3489	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∩ {𝑥}) = ∅ ↔ 𝐵 = (𝐵 ∖ {𝑥}))
25	23, 24	bitr3i 186	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ 𝐵 = (𝐵 ∖ {𝑥}))
26		sseq1 3192	. . . . . . . . . . . . . . . . 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 6795	. . . . . . . . . . . . . 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 2267	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝑥 ∈ suc 𝑦)
35		phplem3g 6873	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ω ∧ 𝑥 ∈ suc 𝑦) → 𝑦 ≈ (suc 𝑦 ∖ {𝑥}))
36	35	ensymd 6800	. . . . . . . . . . . . . 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 6808	. . . . . . . . . . . . 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 6807	. . . . . . . . . . 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 4041	. . . . . . . . 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 2606	. . . . . 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 4617	. . . . . 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 1829	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝐴 ≈ 𝐵))
53	52	3impia 1201	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) → ¬ 𝐴 ≈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∧ w3a 979   = wceq 1363  ∃wex 1502   ∈ wcel 2159   ≠ wne 2359  ∃wrex 2468  Vcvv 2751   ∖ cdif 3140   ∩ cin 3142   ⊆ wss 3143  ∅c0 3436  {csn 3606   class class class wbr 4017  suc csuc 4379  ωcom 4603   ≈ cen 6755   ≼ cdom 6756

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-ral 2472  df-rex 2473  df-rab 2476  df-v 2753  df-sbc 2977  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-br 4018  df-opab 4079  df-tr 4116  df-id 4307  df-iord 4380  df-on 4382  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-er 6552  df-en 6758  df-dom 6759

This theorem is referenced by:  phpelm  6883