Theorem ssfin4 10207

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 4047	. . . . . . . . 9 ⊢ (𝑥 ⊊ 𝐵 → 𝑥 ⊆ 𝐵)
3		simpr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
4	2, 3	sylan9ssr 3944	. . . . . . . 8 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → 𝑥 ⊆ 𝐴)
5		difssd 4086	. . . . . . . 8 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → (𝐴 ∖ 𝐵) ⊆ 𝐴)
6	4, 5	unssd 4141	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴)
7		pssnel 4420	. . . . . . . . 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 3930	. . . . . . . . . 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 4082	. . . . . . . . . . . 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 4102	. . . . . . . . . . . 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 2856	. . . . . . . . . 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 2738	. . . . . . . . 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 1936	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → ¬ (𝑥 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
24		dfpss2 4037	. . . . . . 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 5269	. . . . . . . 8 ⊢ (𝐴 ∈ FinIV → (𝐴 ∖ 𝐵) ∈ V)
29		enrefg 8912	. . . . . . . 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 4195	. . . . . . . . . 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 4323	. . . . . . . . 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 4421	. . . . . . . 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 8973	. . . . . . 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 4431	. . . . . . 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 5119	. . . . 5 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ≈ 𝐴)
44		fin4i 10195	. . . . 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 1937	. 2 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵))
48		ssexg 5263	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ FinIV) → 𝐵 ∈ V)
49	48	ancoms 458	. . 3 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
50		isfin4 10194	. . 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 1541  ∃wex 1780   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3894   ∪ cun 3895   ∩ cin 3896   ⊆ wss 3897   ⊊ wpss 3898  ∅c0 4282   class class class wbr 5093   ≈ cen 8872  Fin^IVcfin4 10177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  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 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-en 8876  df-fin4 10184

This theorem is referenced by:  domfin4  10208