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Theorem ssfin4 9084
Description: Dedekind finite sets have Dedekind finite subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
ssfin4 ((𝐴 ∈ FinIV𝐵𝐴) → 𝐵 ∈ FinIV)

Proof of Theorem ssfin4
Dummy variables 𝑐 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 789 . . . 4 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝐴 ∈ FinIV)
2 pssss 3685 . . . . . . . . 9 (𝑥𝐵𝑥𝐵)
3 simpr 477 . . . . . . . . 9 ((𝐴 ∈ FinIV𝐵𝐴) → 𝐵𝐴)
42, 3sylan9ssr 3601 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → 𝑥𝐴)
5 difssd 3721 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → (𝐴𝐵) ⊆ 𝐴)
64, 5unssd 3772 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → (𝑥 ∪ (𝐴𝐵)) ⊆ 𝐴)
7 pssnel 4016 . . . . . . . . 9 (𝑥𝐵 → ∃𝑐(𝑐𝐵 ∧ ¬ 𝑐𝑥))
87adantl 482 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → ∃𝑐(𝑐𝐵 ∧ ¬ 𝑐𝑥))
9 simpllr 798 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → 𝐵𝐴)
10 simprl 793 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → 𝑐𝐵)
119, 10sseldd 3588 . . . . . . . . . 10 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → 𝑐𝐴)
12 simprr 795 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝑐𝑥)
13 elndif 3717 . . . . . . . . . . . 12 (𝑐𝐵 → ¬ 𝑐 ∈ (𝐴𝐵))
1413ad2antrl 763 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝑐 ∈ (𝐴𝐵))
15 ioran 511 . . . . . . . . . . . 12 (¬ (𝑐𝑥𝑐 ∈ (𝐴𝐵)) ↔ (¬ 𝑐𝑥 ∧ ¬ 𝑐 ∈ (𝐴𝐵)))
16 elun 3736 . . . . . . . . . . . 12 (𝑐 ∈ (𝑥 ∪ (𝐴𝐵)) ↔ (𝑐𝑥𝑐 ∈ (𝐴𝐵)))
1715, 16xchnxbir 323 . . . . . . . . . . 11 𝑐 ∈ (𝑥 ∪ (𝐴𝐵)) ↔ (¬ 𝑐𝑥 ∧ ¬ 𝑐 ∈ (𝐴𝐵)))
1812, 14, 17sylanbrc 697 . . . . . . . . . 10 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝑐 ∈ (𝑥 ∪ (𝐴𝐵)))
19 nelneq2 2723 . . . . . . . . . 10 ((𝑐𝐴 ∧ ¬ 𝑐 ∈ (𝑥 ∪ (𝐴𝐵))) → ¬ 𝐴 = (𝑥 ∪ (𝐴𝐵)))
2011, 18, 19syl2anc 692 . . . . . . . . 9 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝐴 = (𝑥 ∪ (𝐴𝐵)))
21 eqcom 2628 . . . . . . . . 9 (𝐴 = (𝑥 ∪ (𝐴𝐵)) ↔ (𝑥 ∪ (𝐴𝐵)) = 𝐴)
2220, 21sylnib 318 . . . . . . . 8 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ (𝑥 ∪ (𝐴𝐵)) = 𝐴)
238, 22exlimddv 1860 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → ¬ (𝑥 ∪ (𝐴𝐵)) = 𝐴)
24 dfpss2 3675 . . . . . . 7 ((𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴 ↔ ((𝑥 ∪ (𝐴𝐵)) ⊆ 𝐴 ∧ ¬ (𝑥 ∪ (𝐴𝐵)) = 𝐴))
256, 23, 24sylanbrc 697 . . . . . 6 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → (𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴)
2625adantrr 752 . . . . 5 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴)
27 simprr 795 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝑥𝐵)
28 difexg 4773 . . . . . . . 8 (𝐴 ∈ FinIV → (𝐴𝐵) ∈ V)
29 enrefg 7939 . . . . . . . 8 ((𝐴𝐵) ∈ V → (𝐴𝐵) ≈ (𝐴𝐵))
301, 28, 293syl 18 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝐴𝐵) ≈ (𝐴𝐵))
312ad2antrl 763 . . . . . . . . . 10 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝑥𝐵)
32 ssinss1 3824 . . . . . . . . . 10 (𝑥𝐵 → (𝑥𝐴) ⊆ 𝐵)
3331, 32syl 17 . . . . . . . . 9 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥𝐴) ⊆ 𝐵)
34 inssdif0 3926 . . . . . . . . 9 ((𝑥𝐴) ⊆ 𝐵 ↔ (𝑥 ∩ (𝐴𝐵)) = ∅)
3533, 34sylib 208 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∩ (𝐴𝐵)) = ∅)
36 disjdif 4017 . . . . . . . 8 (𝐵 ∩ (𝐴𝐵)) = ∅
3735, 36jctir 560 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → ((𝑥 ∩ (𝐴𝐵)) = ∅ ∧ (𝐵 ∩ (𝐴𝐵)) = ∅))
38 unen 7992 . . . . . . 7 (((𝑥𝐵 ∧ (𝐴𝐵) ≈ (𝐴𝐵)) ∧ ((𝑥 ∩ (𝐴𝐵)) = ∅ ∧ (𝐵 ∩ (𝐴𝐵)) = ∅)) → (𝑥 ∪ (𝐴𝐵)) ≈ (𝐵 ∪ (𝐴𝐵)))
3927, 30, 37, 38syl21anc 1322 . . . . . 6 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∪ (𝐴𝐵)) ≈ (𝐵 ∪ (𝐴𝐵)))
40 simplr 791 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝐵𝐴)
41 undif 4026 . . . . . . 7 (𝐵𝐴 ↔ (𝐵 ∪ (𝐴𝐵)) = 𝐴)
4240, 41sylib 208 . . . . . 6 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝐵 ∪ (𝐴𝐵)) = 𝐴)
4339, 42breqtrd 4644 . . . . 5 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∪ (𝐴𝐵)) ≈ 𝐴)
44 fin4i 9072 . . . . 5 (((𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴 ∧ (𝑥 ∪ (𝐴𝐵)) ≈ 𝐴) → ¬ 𝐴 ∈ FinIV)
4526, 43, 44syl2anc 692 . . . 4 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → ¬ 𝐴 ∈ FinIV)
461, 45pm2.65da 599 . . 3 ((𝐴 ∈ FinIV𝐵𝐴) → ¬ (𝑥𝐵𝑥𝐵))
4746nexdv 1861 . 2 ((𝐴 ∈ FinIV𝐵𝐴) → ¬ ∃𝑥(𝑥𝐵𝑥𝐵))
48 ssexg 4769 . . . 4 ((𝐵𝐴𝐴 ∈ FinIV) → 𝐵 ∈ V)
4948ancoms 469 . . 3 ((𝐴 ∈ FinIV𝐵𝐴) → 𝐵 ∈ V)
50 isfin4 9071 . . 3 (𝐵 ∈ V → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐵)))
5149, 50syl 17 . 2 ((𝐴 ∈ FinIV𝐵𝐴) → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐵)))
5247, 51mpbird 247 1 ((𝐴 ∈ FinIV𝐵𝐴) → 𝐵 ∈ FinIV)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3189  cdif 3556  cun 3557  cin 3558  wss 3559  wpss 3560  c0 3896   class class class wbr 4618  cen 7904  FinIVcfin4 9054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-en 7908  df-fin4 9061
This theorem is referenced by:  domfin4  9085
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