Theorem phpm 7157

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 7143 through phplem4 7146, nneneq 7148, 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 3351	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐴)
3		ne0i 3528	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
4	2, 3	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝐴 ≠ ∅)
5	4	neneqd 2441	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝐴 = ∅)
6	5	ad2antlr 493	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝐴 = ∅) → ¬ 𝐴 = ∅)
7	1, 6	pm2.21dd 629	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝐴 = ∅) → ¬ 𝐴 ≈ 𝐵)
8		php5dom 7154	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → ¬ suc 𝑦 ≼ 𝑦)
9	8	ad2antlr 493	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → ¬ suc 𝑦 ≼ 𝑦)
10		simplr 533	. . . . . . . . . 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 2824	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
13	12	sucex 4641	. . . . . . . . . . . . . . 15 ⊢ suc 𝑦 ∈ V
14		difss 3355	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑦 ∖ {𝑥}) ⊆ suc 𝑦
15	13, 14	ssexi 4266	. . . . . . . . . . . . . 14 ⊢ (suc 𝑦 ∖ {𝑥}) ∈ V
16		eldifn 3352	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵)
17	16	ad3antlr 497	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → ¬ 𝑥 ∈ 𝐵)
18		simpllr 540	. . . . . . . . . . . . . . . . 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 3286	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝐵 ⊆ suc 𝑦)
22		ssdif 3364	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ⊆ suc 𝑦 → (𝐵 ∖ {𝑥}) ⊆ (suc 𝑦 ∖ {𝑥}))
23		disjsn 3767	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝐵)
24		disj3 3576	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∩ {𝑥}) = ∅ ↔ 𝐵 = (𝐵 ∖ {𝑥}))
25	23, 24	bitr3i 186	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ 𝐵 = (𝐵 ∖ {𝑥}))
26		sseq1 3271	. . . . . . . . . . . . . . . . 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 7055	. . . . . . . . . . . . . 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 533	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝑦 ∈ ω)
33	2	ad3antlr 497	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝑥 ∈ 𝐴)
34	33, 20	eleqtrd 2317	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → 𝑥 ∈ suc 𝑦)
35		phplem3g 7147	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ω ∧ 𝑥 ∈ suc 𝑦) → 𝑦 ≈ (suc 𝑦 ∖ {𝑥}))
36	35	ensymd 7060	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ω ∧ 𝑥 ∈ suc 𝑦) → (suc 𝑦 ∖ {𝑥}) ≈ 𝑦)
37	32, 34, 36	syl2anc 415	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) → (suc 𝑦 ∖ {𝑥}) ≈ 𝑦)
38		domentr 7068	. . . . . . . . . . . . 13 ⊢ ((𝐵 ≼ (suc 𝑦 ∖ {𝑥}) ∧ (suc 𝑦 ∖ {𝑥}) ≈ 𝑦) → 𝐵 ≼ 𝑦)
39	31, 37, 38	syl2anc 415	. . . . . . . . . . . 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 7067	. . . . . . . . . . 11 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≼ 𝑦) → 𝐴 ≼ 𝑦)
42	11, 40, 41	syl2anc 415	. . . . . . . . . 10 ⊢ ((((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴 ≈ 𝐵) → 𝐴 ≼ 𝑦)
43	10, 42	eqbrtrrd 4149	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) ∧ 𝑦 ∈ ω) ∧ 𝐴 = suc 𝑦) ∧ 𝐴 ≈ 𝐵) → suc 𝑦 ≼ 𝑦)
44	9, 43	mtand 675	. . . . . . . 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 2668	. . . . . 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 4746	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑦 ∈ ω 𝐴 = suc 𝑦))
49	48	ad2antrr 492	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ 𝐵)) → (𝐴 = ∅ ∨ ∃𝑦 ∈ ω 𝐴 = suc 𝑦))
50	7, 47, 49	mpjaodan 810	. . . 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 1872	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝐴 ≈ 𝐵))
53	52	3impia 1231	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) → ¬ 𝐴 ≈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209   ≠ wne 2420  ∃wrex 2529  Vcvv 2821   ∖ cdif 3217   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520  {csn 3705   class class class wbr 4125  suc csuc 4505  ωcom 4732   ≈ cen 7010   ≼ cdom 7011

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-er 6797  df-en 7013  df-dom 7014

This theorem is referenced by:  phpelm  7158