Theorem ssfin4 10381

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 779	. . . 4 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝐴 ∈ FinIV)
2		pssss 4046	. . . . . . . . 9 ⊢ (𝑥 ⊊ 𝐵 → 𝑥 ⊆ 𝐵)
3		simpr 490	. . . . . . . . 9 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
4	2, 3	sylan9ssr 3945	. . . . . . . 8 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → 𝑥 ⊆ 𝐴)
5		difssd 4084	. . . . . . . 8 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → (𝐴 ∖ 𝐵) ⊆ 𝐴)
6	4, 5	unssd 4138	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴)
7		pssnel 4424	. . . . . . . . 9 ⊢ (𝑥 ⊊ 𝐵 → ∃𝑐(𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥))
8	7	adantl 487	. . . . . . . 8 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → ∃𝑐(𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥))
9		simpllr 788	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → 𝐵 ⊆ 𝐴)
10		simprl 783	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → 𝑐 ∈ 𝐵)
11	9, 10	sseldd 3932	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → 𝑐 ∈ 𝐴)
12		simprr 785	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → ¬ 𝑐 ∈ 𝑥)
13		elndif 4080	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝐵 → ¬ 𝑐 ∈ (𝐴 ∖ 𝐵))
14	13	ad2antrl 741	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → ¬ 𝑐 ∈ (𝐴 ∖ 𝐵))
15		ioran 999	. . . . . . . . . . . 12 ⊢ (¬ (𝑐 ∈ 𝑥 ∨ 𝑐 ∈ (𝐴 ∖ 𝐵)) ↔ (¬ 𝑐 ∈ 𝑥 ∧ ¬ 𝑐 ∈ (𝐴 ∖ 𝐵)))
16		elun 4100	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (𝑥 ∪ (𝐴 ∖ 𝐵)) ↔ (𝑐 ∈ 𝑥 ∨ 𝑐 ∈ (𝐴 ∖ 𝐵)))
17	15, 16	xchnxbir 336	. . . . . . . . . . 11 ⊢ (¬ 𝑐 ∈ (𝑥 ∪ (𝐴 ∖ 𝐵)) ↔ (¬ 𝑐 ∈ 𝑥 ∧ ¬ 𝑐 ∈ (𝐴 ∖ 𝐵)))
18	12, 14, 17	sylanbrc 595	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → ¬ 𝑐 ∈ (𝑥 ∪ (𝐴 ∖ 𝐵)))
19		nelneq2 2886	. . . . . . . . . 10 ⊢ ((𝑐 ∈ 𝐴 ∧ ¬ 𝑐 ∈ (𝑥 ∪ (𝐴 ∖ 𝐵))) → ¬ 𝐴 = (𝑥 ∪ (𝐴 ∖ 𝐵)))
20	11, 18, 19	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → ¬ 𝐴 = (𝑥 ∪ (𝐴 ∖ 𝐵)))
21		eqcom 2768	. . . . . . . . 9 ⊢ (𝐴 = (𝑥 ∪ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
22	20, 21	sylnib 331	. . . . . . . 8 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → ¬ (𝑥 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
23	8, 22	exlimddv 1968	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → ¬ (𝑥 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
24		dfpss2 4036	. . . . . . 7 ⊢ ((𝑥 ∪ (𝐴 ∖ 𝐵)) ⊊ 𝐴 ↔ ((𝑥 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴 ∧ ¬ (𝑥 ∪ (𝐴 ∖ 𝐵)) = 𝐴))
25	6, 23, 24	sylanbrc 595	. . . . . 6 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ⊊ 𝐴)
26	25	adantrr 730	. . . . 5 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ⊊ 𝐴)
27		simprr 785	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝑥 ≈ 𝐵)
28		difexg 5291	. . . . . . . 8 ⊢ (𝐴 ∈ FinIV → (𝐴 ∖ 𝐵) ∈ V)
29		enrefg 9004	. . . . . . . 8 ⊢ ((𝐴 ∖ 𝐵) ∈ V → (𝐴 ∖ 𝐵) ≈ (𝐴 ∖ 𝐵))
30	1, 28, 29	3syl 19	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝐴 ∖ 𝐵) ≈ (𝐴 ∖ 𝐵))
31	2	ad2antrl 741	. . . . . . . . . 10 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝑥 ⊆ 𝐵)
32		ssinss1 4191	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ∩ 𝐴) ⊆ 𝐵)
33	31, 32	syl 18	. . . . . . . . 9 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑥 ∩ 𝐴) ⊆ 𝐵)
34		inssdif0 4322	. . . . . . . . 9 ⊢ ((𝑥 ∩ 𝐴) ⊆ 𝐵 ↔ (𝑥 ∩ (𝐴 ∖ 𝐵)) = ∅)
35	33, 34	sylib 221	. . . . . . . 8 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑥 ∩ (𝐴 ∖ 𝐵)) = ∅)
36		disjdif 4426	. . . . . . . 8 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅
37	35, 36	jctir 530	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → ((𝑥 ∩ (𝐴 ∖ 𝐵)) = ∅ ∧ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅))
38		unen 9066	. . . . . . 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 851	. . . . . 6 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ≈ (𝐵 ∪ (𝐴 ∖ 𝐵)))
40		simplr 781	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝐵 ⊆ 𝐴)
41		undif 4438	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
42	40, 41	sylib 221	. . . . . 6 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
43	39, 42	breqtrd 5131	. . . . 5 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ≈ 𝐴)
44		fin4i 10369	. . . . 5 ⊢ (((𝑥 ∪ (𝐴 ∖ 𝐵)) ⊊ 𝐴 ∧ (𝑥 ∪ (𝐴 ∖ 𝐵)) ≈ 𝐴) → ¬ 𝐴 ∈ FinIV)
45	26, 43, 44	syl2anc 596	. . . 4 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → ¬ 𝐴 ∈ FinIV)
46	1, 45	pm2.65da 829	. . 3 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → ¬ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵))
47	46	nexdv 1969	. 2 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵))
48		ssexg 5281	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ FinIV) → 𝐵 ∈ V)
49	48	ancoms 464	. . 3 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
50		isfin4 10368	. . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
51	49, 50	syl 18	. 2 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
52	47, 51	mpbird 260	1 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ FinIV)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3451   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899   ⊊ wpss 3900  ∅c0 4279   class class class wbr 5103   ≈ cen 8963  Fin^IVcfin4 10351

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-en 8967  df-fin4 10358

This theorem is used by:  domfin4  10382