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Theorem ssfin4 10167
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 ∈ FinIV𝐵𝐴) → 𝐵 ∈ FinIV)

Proof of Theorem ssfin4
Dummy variables 𝑐 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 764 . . . 4 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝐴 ∈ FinIV)
2 pssss 4042 . . . . . . . . 9 (𝑥𝐵𝑥𝐵)
3 simpr 485 . . . . . . . . 9 ((𝐴 ∈ FinIV𝐵𝐴) → 𝐵𝐴)
42, 3sylan9ssr 3946 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → 𝑥𝐴)
5 difssd 4079 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → (𝐴𝐵) ⊆ 𝐴)
64, 5unssd 4133 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → (𝑥 ∪ (𝐴𝐵)) ⊆ 𝐴)
7 pssnel 4417 . . . . . . . . 9 (𝑥𝐵 → ∃𝑐(𝑐𝐵 ∧ ¬ 𝑐𝑥))
87adantl 482 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → ∃𝑐(𝑐𝐵 ∧ ¬ 𝑐𝑥))
9 simpllr 773 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → 𝐵𝐴)
10 simprl 768 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → 𝑐𝐵)
119, 10sseldd 3933 . . . . . . . . . 10 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → 𝑐𝐴)
12 simprr 770 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝑐𝑥)
13 elndif 4075 . . . . . . . . . . . 12 (𝑐𝐵 → ¬ 𝑐 ∈ (𝐴𝐵))
1413ad2antrl 725 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝑐 ∈ (𝐴𝐵))
15 ioran 981 . . . . . . . . . . . 12 (¬ (𝑐𝑥𝑐 ∈ (𝐴𝐵)) ↔ (¬ 𝑐𝑥 ∧ ¬ 𝑐 ∈ (𝐴𝐵)))
16 elun 4095 . . . . . . . . . . . 12 (𝑐 ∈ (𝑥 ∪ (𝐴𝐵)) ↔ (𝑐𝑥𝑐 ∈ (𝐴𝐵)))
1715, 16xchnxbir 332 . . . . . . . . . . 11 𝑐 ∈ (𝑥 ∪ (𝐴𝐵)) ↔ (¬ 𝑐𝑥 ∧ ¬ 𝑐 ∈ (𝐴𝐵)))
1812, 14, 17sylanbrc 583 . . . . . . . . . 10 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝑐 ∈ (𝑥 ∪ (𝐴𝐵)))
19 nelneq2 2862 . . . . . . . . . 10 ((𝑐𝐴 ∧ ¬ 𝑐 ∈ (𝑥 ∪ (𝐴𝐵))) → ¬ 𝐴 = (𝑥 ∪ (𝐴𝐵)))
2011, 18, 19syl2anc 584 . . . . . . . . 9 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝐴 = (𝑥 ∪ (𝐴𝐵)))
21 eqcom 2743 . . . . . . . . 9 (𝐴 = (𝑥 ∪ (𝐴𝐵)) ↔ (𝑥 ∪ (𝐴𝐵)) = 𝐴)
2220, 21sylnib 327 . . . . . . . 8 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ (𝑥 ∪ (𝐴𝐵)) = 𝐴)
238, 22exlimddv 1937 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → ¬ (𝑥 ∪ (𝐴𝐵)) = 𝐴)
24 dfpss2 4032 . . . . . . 7 ((𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴 ↔ ((𝑥 ∪ (𝐴𝐵)) ⊆ 𝐴 ∧ ¬ (𝑥 ∪ (𝐴𝐵)) = 𝐴))
256, 23, 24sylanbrc 583 . . . . . 6 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → (𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴)
2625adantrr 714 . . . . 5 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴)
27 simprr 770 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝑥𝐵)
28 difexg 5271 . . . . . . . 8 (𝐴 ∈ FinIV → (𝐴𝐵) ∈ V)
29 enrefg 8845 . . . . . . . 8 ((𝐴𝐵) ∈ V → (𝐴𝐵) ≈ (𝐴𝐵))
301, 28, 293syl 18 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝐴𝐵) ≈ (𝐴𝐵))
312ad2antrl 725 . . . . . . . . . 10 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝑥𝐵)
32 ssinss1 4184 . . . . . . . . . 10 (𝑥𝐵 → (𝑥𝐴) ⊆ 𝐵)
3331, 32syl 17 . . . . . . . . 9 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥𝐴) ⊆ 𝐵)
34 inssdif0 4316 . . . . . . . . 9 ((𝑥𝐴) ⊆ 𝐵 ↔ (𝑥 ∩ (𝐴𝐵)) = ∅)
3533, 34sylib 217 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∩ (𝐴𝐵)) = ∅)
36 disjdif 4418 . . . . . . . 8 (𝐵 ∩ (𝐴𝐵)) = ∅
3735, 36jctir 521 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → ((𝑥 ∩ (𝐴𝐵)) = ∅ ∧ (𝐵 ∩ (𝐴𝐵)) = ∅))
38 unen 8911 . . . . . . 7 (((𝑥𝐵 ∧ (𝐴𝐵) ≈ (𝐴𝐵)) ∧ ((𝑥 ∩ (𝐴𝐵)) = ∅ ∧ (𝐵 ∩ (𝐴𝐵)) = ∅)) → (𝑥 ∪ (𝐴𝐵)) ≈ (𝐵 ∪ (𝐴𝐵)))
3927, 30, 37, 38syl21anc 835 . . . . . 6 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∪ (𝐴𝐵)) ≈ (𝐵 ∪ (𝐴𝐵)))
40 simplr 766 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝐵𝐴)
41 undif 4428 . . . . . . 7 (𝐵𝐴 ↔ (𝐵 ∪ (𝐴𝐵)) = 𝐴)
4240, 41sylib 217 . . . . . 6 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝐵 ∪ (𝐴𝐵)) = 𝐴)
4339, 42breqtrd 5118 . . . . 5 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∪ (𝐴𝐵)) ≈ 𝐴)
44 fin4i 10155 . . . . 5 (((𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴 ∧ (𝑥 ∪ (𝐴𝐵)) ≈ 𝐴) → ¬ 𝐴 ∈ FinIV)
4526, 43, 44syl2anc 584 . . . 4 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → ¬ 𝐴 ∈ FinIV)
461, 45pm2.65da 814 . . 3 ((𝐴 ∈ FinIV𝐵𝐴) → ¬ (𝑥𝐵𝑥𝐵))
4746nexdv 1938 . 2 ((𝐴 ∈ FinIV𝐵𝐴) → ¬ ∃𝑥(𝑥𝐵𝑥𝐵))
48 ssexg 5267 . . . 4 ((𝐵𝐴𝐴 ∈ FinIV) → 𝐵 ∈ V)
4948ancoms 459 . . 3 ((𝐴 ∈ FinIV𝐵𝐴) → 𝐵 ∈ V)
50 isfin4 10154 . . 3 (𝐵 ∈ V → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐵)))
5149, 50syl 17 . 2 ((𝐴 ∈ FinIV𝐵𝐴) → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐵)))
5247, 51mpbird 256 1 ((𝐴 ∈ FinIV𝐵𝐴) → 𝐵 ∈ FinIV)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844   = wceq 1540  wex 1780  wcel 2105  Vcvv 3441  cdif 3895  cun 3896  cin 3897  wss 3898  wpss 3899  c0 4269   class class class wbr 5092  cen 8801  FinIVcfin4 10137
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-en 8805  df-fin4 10144
This theorem is referenced by:  domfin4  10168
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