Theorem ssfin4 10290

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 778	. . . 4 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝐴 ∈ FinIV)
2		pssss 4060	. . . . . . . . 9 ⊢ (𝑥 ⊊ 𝐵 → 𝑥 ⊆ 𝐵)
3		simpr 489	. . . . . . . . 9 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
4	2, 3	sylan9ssr 3959	. . . . . . . 8 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → 𝑥 ⊆ 𝐴)
5		difssd 4099	. . . . . . . 8 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → (𝐴 ∖ 𝐵) ⊆ 𝐴)
6	4, 5	unssd 4153	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴)
7		pssnel 4434	. . . . . . . . 9 ⊢ (𝑥 ⊊ 𝐵 → ∃𝑐(𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥))
8	7	adantl 486	. . . . . . . 8 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → ∃𝑐(𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥))
9		simpllr 787	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → 𝐵 ⊆ 𝐴)
10		simprl 782	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → 𝑐 ∈ 𝐵)
11	9, 10	sseldd 3946	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → 𝑐 ∈ 𝐴)
12		simprr 784	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → ¬ 𝑐 ∈ 𝑥)
13		elndif 4095	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝐵 → ¬ 𝑐 ∈ (𝐴 ∖ 𝐵))
14	13	ad2antrl 740	. . . . . . . . . . 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 4115	. . . . . . . . . . . 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 594	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → ¬ 𝑐 ∈ (𝑥 ∪ (𝐴 ∖ 𝐵)))
19		nelneq2 2894	. . . . . . . . . 10 ⊢ ((𝑐 ∈ 𝐴 ∧ ¬ 𝑐 ∈ (𝑥 ∪ (𝐴 ∖ 𝐵))) → ¬ 𝐴 = (𝑥 ∪ (𝐴 ∖ 𝐵)))
20	11, 18, 19	syl2anc 595	. . . . . . . . 9 ⊢ ((((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) ∧ (𝑐 ∈ 𝐵 ∧ ¬ 𝑐 ∈ 𝑥)) → ¬ 𝐴 = (𝑥 ∪ (𝐴 ∖ 𝐵)))
21		eqcom 2776	. . . . . . . . 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 1962	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → ¬ (𝑥 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
24		dfpss2 4050	. . . . . . 7 ⊢ ((𝑥 ∪ (𝐴 ∖ 𝐵)) ⊊ 𝐴 ↔ ((𝑥 ∪ (𝐴 ∖ 𝐵)) ⊆ 𝐴 ∧ ¬ (𝑥 ∪ (𝐴 ∖ 𝐵)) = 𝐴))
25	6, 23, 24	sylanbrc 594	. . . . . 6 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ⊊ 𝐵) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ⊊ 𝐴)
26	25	adantrr 729	. . . . 5 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ⊊ 𝐴)
27		simprr 784	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝑥 ≈ 𝐵)
28		difexg 5297	. . . . . . . 8 ⊢ (𝐴 ∈ FinIV → (𝐴 ∖ 𝐵) ∈ V)
29		enrefg 8977	. . . . . . . 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 740	. . . . . . . . . 10 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝑥 ⊆ 𝐵)
32		ssinss1 4206	. . . . . . . . . 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 4336	. . . . . . . . 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 4435	. . . . . . . 8 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅
37	35, 36	jctir 529	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → ((𝑥 ∩ (𝐴 ∖ 𝐵)) = ∅ ∧ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅))
38		unen 9038	. . . . . . 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 850	. . . . . 6 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ≈ (𝐵 ∪ (𝐴 ∖ 𝐵)))
40		simplr 780	. . . . . . 7 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝐵 ⊆ 𝐴)
41		undif 4445	. . . . . . 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 5138	. . . . 5 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑥 ∪ (𝐴 ∖ 𝐵)) ≈ 𝐴)
44		fin4i 10278	. . . . 5 ⊢ (((𝑥 ∪ (𝐴 ∖ 𝐵)) ⊊ 𝐴 ∧ (𝑥 ∪ (𝐴 ∖ 𝐵)) ≈ 𝐴) → ¬ 𝐴 ∈ FinIV)
45	26, 43, 44	syl2anc 595	. . . 4 ⊢ (((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → ¬ 𝐴 ∈ FinIV)
46	1, 45	pm2.65da 828	. . 3 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → ¬ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵))
47	46	nexdv 1963	. 2 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵))
48		ssexg 5291	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ FinIV) → 𝐵 ∈ V)
49	48	ancoms 463	. . 3 ⊢ ((𝐴 ∈ FinIV ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
50		isfin4 10277	. . 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)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3463   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913   ⊊ wpss 3914  ∅c0 4294   class class class wbr 5110   ≈ cen 8936  Fin^IVcfin4 10260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-en 8940  df-fin4 10267

This theorem is referenced by:  domfin4  10291