Theorem ssfin4 10204

Description: Dedekind finite sets have Dedekind finite subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Revised by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
ssfin4	⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ FinIV)

Proof of Theorem ssfin4

Dummy variables 𝑐 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 766	. . . 4 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝐴 ∈ FinIV)
2		pssss 4049	. . . . . . . . 9 ⊢ (𝑥 ⊊ 𝐵 → 𝑥 ⊆ 𝐵)
3		simpr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
4	2, 3	sylan9ssr 3950	. . . . . . . 8 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → 𝑥 ⊆ 𝐴)
5		difssd 4088	. . . . . . . 8 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → (𝐴 ∖ 𝐵) ⊆ 𝐴)
6	4, 5	unssd 4143	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴)
7		pssnel 4422	. . . . . . . . 9 ⊢ (𝑥 ⊊ 𝐵 → ∃𝑐(𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥))
8	7	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → ∃𝑐(𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥))
9		simpllr 775	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → 𝐵 ⊆ 𝐴)
10		simprl 770	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → 𝑐 ∈ 𝐵)
11	9, 10	sseldd 3936	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → 𝑐 ∈ 𝐴)
12		simprr 772	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → ¬ 𝑐 ∈ 𝑥)
13		elndif 4084	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝐵 → ¬ 𝑐 ∈ (𝐴 ∖ 𝐵))
14	13	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → ¬ 𝑐 ∈ (𝐴 ∖ 𝐵))
15		ioran 985	. . . . . . . . . . . 12 ⊢ (¬ (𝑐 ∈ 𝑥 ∨ 𝑐 ∈ (𝐴 ∖ 𝐵)) ↔ (¬ 𝑐 ∈ 𝑥 ∧ ¬ 𝑐 ∈ (𝐴 ∖ 𝐵)))
16		elun 4104	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (𝑥 ∪ (𝐴 ∖ 𝐵)) ↔ (𝑐 ∈ 𝑥 ∨ 𝑐 ∈ (𝐴 ∖ 𝐵)))
17	15, 16	xchnxbir 333	. . . . . . . . . . 11 ⊢ (¬ 𝑐 ∈ (𝑥 ∪ (𝐴 ∖ 𝐵)) ↔ (¬ 𝑐 ∈ 𝑥 ∧ ¬ 𝑐 ∈ (𝐴 ∖ 𝐵)))
18	12, 14, 17	sylanbrc 583	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → ¬ 𝑐 ∈ (𝑥 ∪ (𝐴 ∖ 𝐵)))
19		nelneq2 2853	. . . . . . . . . 10 ⊢ ((𝑐 ∈ 𝐴 ∧ ¬ 𝑐 ∈ (𝑥 ∪ (𝐴 ∖ 𝐵))) → ¬ 𝐴 = (𝑥 ∪ (𝐴 ∖ 𝐵)))
20	11, 18, 19	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → ¬ 𝐴 = (𝑥 ∪ (𝐴 ∖ 𝐵)))
21		eqcom 2736	. . . . . . . . 9 ⊢ (𝐴 = (𝑥 ∪ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
22	20, 21	sylnib 328	. . . . . . . 8 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → ¬ (𝑥 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
23	8, 22	exlimddv 1935	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → ¬ (𝑥 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
24		dfpss2 4039	. . . . . . 7 ⊢ ((𝑥 ∪ (𝐴 ∖ 𝐵)) ⊊ 𝐴 ↔ ((𝑥 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴 ∧ ¬ (𝑥 ∪ (𝐴 ∖ 𝐵)) = 𝐴))
25	6, 23, 24	sylanbrc 583	. . . . . 6 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ⊊ 𝐴)
26	25	adantrr 717	. . . . 5 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ⊊ 𝐴)
27		simprr 772	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝑥 ≈ 𝐵)
28		difexg 5268	. . . . . . . 8 ⊢ (𝐴 ∈ FinIV → (𝐴 ∖ 𝐵) ∈ V)
29		enrefg 8909	. . . . . . . 8 ⊢ ((𝐴 ∖ 𝐵) ∈ V → (𝐴 ∖ 𝐵) ≈ (𝐴 ∖ 𝐵))
30	1, 28, 29	3syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝐴 ∖ 𝐵) ≈ (𝐴 ∖ 𝐵))
31	2	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝑥 ⊆ 𝐵)
32		ssinss1 4197	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∩ 𝐴) ⊆ 𝐵)
33	31, 32	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑥 ∩ 𝐴) ⊆ 𝐵)
34		inssdif0 4325	. . . . . . . . 9 ⊢ ((𝑥 ∩ 𝐴) ⊆ 𝐵 ↔ (𝑥 ∩ (𝐴 ∖ 𝐵)) = ∅)
35	33, 34	sylib 218	. . . . . . . 8 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑥 ∩ (𝐴 ∖ 𝐵)) = ∅)
36		disjdif 4423	. . . . . . . 8 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅
37	35, 36	jctir 520	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → ((𝑥 ∩ (𝐴 ∖ 𝐵)) = ∅ ∧ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅))
38		unen 8971	. . . . . . 7 ⊢ (((𝑥 ≈ 𝐵 ∧ (𝐴 ∖ 𝐵) ≈ (𝐴 ∖ 𝐵)) ∧ ((𝑥 ∩ (𝐴 ∖ 𝐵)) = ∅ ∧ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅)) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ≈ (𝐵 ∪ (𝐴 ∖ 𝐵)))
39	27, 30, 37, 38	syl21anc 837	. . . . . 6 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ≈ (𝐵 ∪ (𝐴 ∖ 𝐵)))
40		simplr 768	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝐵 ⊆ 𝐴)
41		undif 4433	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
42	40, 41	sylib 218	. . . . . 6 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
43	39, 42	breqtrd 5118	. . . . 5 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ≈ 𝐴)
44		fin4i 10192	. . . . 5 ⊢ (((𝑥 ∪ (𝐴 ∖ 𝐵)) ⊊ 𝐴 ∧ (𝑥 ∪ (𝐴 ∖ 𝐵)) ≈ 𝐴) → ¬ 𝐴 ∈ FinIV)
45	26, 43, 44	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → ¬ 𝐴 ∈ FinIV)
46	1, 45	pm2.65da 816	. . 3 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → ¬ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵))
47	46	nexdv 1936	. 2 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵))
48		ssexg 5262	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ FinIV) → 𝐵 ∈ V)
49	48	ancoms 458	. . 3 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
50		isfin4 10191	. . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
51	49, 50	syl 17	. 2 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
52	47, 51	mpbird 257	1 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ FinIV)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3436   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903   ⊊ wpss 3904  ∅c0 4284   class class class wbr 5092   ≈ cen 8869  Fin^IVcfin4 10174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-en 8873  df-fin4 10181

This theorem is referenced by:  domfin4  10205