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Theorem ssfin4 10348
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 767 . . . 4 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝐴 ∈ FinIV)
2 pssss 4108 . . . . . . . . 9 (𝑥𝐵𝑥𝐵)
3 simpr 484 . . . . . . . . 9 ((𝐴 ∈ FinIV𝐵𝐴) → 𝐵𝐴)
42, 3sylan9ssr 4010 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → 𝑥𝐴)
5 difssd 4147 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → (𝐴𝐵) ⊆ 𝐴)
64, 5unssd 4202 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → (𝑥 ∪ (𝐴𝐵)) ⊆ 𝐴)
7 pssnel 4477 . . . . . . . . 9 (𝑥𝐵 → ∃𝑐(𝑐𝐵 ∧ ¬ 𝑐𝑥))
87adantl 481 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → ∃𝑐(𝑐𝐵 ∧ ¬ 𝑐𝑥))
9 simpllr 776 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → 𝐵𝐴)
10 simprl 771 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → 𝑐𝐵)
119, 10sseldd 3996 . . . . . . . . . 10 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → 𝑐𝐴)
12 simprr 773 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝑐𝑥)
13 elndif 4143 . . . . . . . . . . . 12 (𝑐𝐵 → ¬ 𝑐 ∈ (𝐴𝐵))
1413ad2antrl 728 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝑐 ∈ (𝐴𝐵))
15 ioran 985 . . . . . . . . . . . 12 (¬ (𝑐𝑥𝑐 ∈ (𝐴𝐵)) ↔ (¬ 𝑐𝑥 ∧ ¬ 𝑐 ∈ (𝐴𝐵)))
16 elun 4163 . . . . . . . . . . . 12 (𝑐 ∈ (𝑥 ∪ (𝐴𝐵)) ↔ (𝑐𝑥𝑐 ∈ (𝐴𝐵)))
1715, 16xchnxbir 333 . . . . . . . . . . 11 𝑐 ∈ (𝑥 ∪ (𝐴𝐵)) ↔ (¬ 𝑐𝑥 ∧ ¬ 𝑐 ∈ (𝐴𝐵)))
1812, 14, 17sylanbrc 583 . . . . . . . . . 10 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝑐 ∈ (𝑥 ∪ (𝐴𝐵)))
19 nelneq2 2864 . . . . . . . . . 10 ((𝑐𝐴 ∧ ¬ 𝑐 ∈ (𝑥 ∪ (𝐴𝐵))) → ¬ 𝐴 = (𝑥 ∪ (𝐴𝐵)))
2011, 18, 19syl2anc 584 . . . . . . . . 9 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝐴 = (𝑥 ∪ (𝐴𝐵)))
21 eqcom 2742 . . . . . . . . 9 (𝐴 = (𝑥 ∪ (𝐴𝐵)) ↔ (𝑥 ∪ (𝐴𝐵)) = 𝐴)
2220, 21sylnib 328 . . . . . . . 8 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ (𝑥 ∪ (𝐴𝐵)) = 𝐴)
238, 22exlimddv 1933 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → ¬ (𝑥 ∪ (𝐴𝐵)) = 𝐴)
24 dfpss2 4098 . . . . . . 7 ((𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴 ↔ ((𝑥 ∪ (𝐴𝐵)) ⊆ 𝐴 ∧ ¬ (𝑥 ∪ (𝐴𝐵)) = 𝐴))
256, 23, 24sylanbrc 583 . . . . . 6 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → (𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴)
2625adantrr 717 . . . . 5 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴)
27 simprr 773 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝑥𝐵)
28 difexg 5335 . . . . . . . 8 (𝐴 ∈ FinIV → (𝐴𝐵) ∈ V)
29 enrefg 9023 . . . . . . . 8 ((𝐴𝐵) ∈ V → (𝐴𝐵) ≈ (𝐴𝐵))
301, 28, 293syl 18 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝐴𝐵) ≈ (𝐴𝐵))
312ad2antrl 728 . . . . . . . . . 10 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝑥𝐵)
32 ssinss1 4254 . . . . . . . . . 10 (𝑥𝐵 → (𝑥𝐴) ⊆ 𝐵)
3331, 32syl 17 . . . . . . . . 9 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥𝐴) ⊆ 𝐵)
34 inssdif0 4380 . . . . . . . . 9 ((𝑥𝐴) ⊆ 𝐵 ↔ (𝑥 ∩ (𝐴𝐵)) = ∅)
3533, 34sylib 218 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∩ (𝐴𝐵)) = ∅)
36 disjdif 4478 . . . . . . . 8 (𝐵 ∩ (𝐴𝐵)) = ∅
3735, 36jctir 520 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → ((𝑥 ∩ (𝐴𝐵)) = ∅ ∧ (𝐵 ∩ (𝐴𝐵)) = ∅))
38 unen 9085 . . . . . . 7 (((𝑥𝐵 ∧ (𝐴𝐵) ≈ (𝐴𝐵)) ∧ ((𝑥 ∩ (𝐴𝐵)) = ∅ ∧ (𝐵 ∩ (𝐴𝐵)) = ∅)) → (𝑥 ∪ (𝐴𝐵)) ≈ (𝐵 ∪ (𝐴𝐵)))
3927, 30, 37, 38syl21anc 838 . . . . . 6 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∪ (𝐴𝐵)) ≈ (𝐵 ∪ (𝐴𝐵)))
40 simplr 769 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝐵𝐴)
41 undif 4488 . . . . . . 7 (𝐵𝐴 ↔ (𝐵 ∪ (𝐴𝐵)) = 𝐴)
4240, 41sylib 218 . . . . . 6 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝐵 ∪ (𝐴𝐵)) = 𝐴)
4339, 42breqtrd 5174 . . . . 5 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∪ (𝐴𝐵)) ≈ 𝐴)
44 fin4i 10336 . . . . 5 (((𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴 ∧ (𝑥 ∪ (𝐴𝐵)) ≈ 𝐴) → ¬ 𝐴 ∈ FinIV)
4526, 43, 44syl2anc 584 . . . 4 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → ¬ 𝐴 ∈ FinIV)
461, 45pm2.65da 817 . . 3 ((𝐴 ∈ FinIV𝐵𝐴) → ¬ (𝑥𝐵𝑥𝐵))
4746nexdv 1934 . 2 ((𝐴 ∈ FinIV𝐵𝐴) → ¬ ∃𝑥(𝑥𝐵𝑥𝐵))
48 ssexg 5329 . . . 4 ((𝐵𝐴𝐴 ∈ FinIV) → 𝐵 ∈ V)
4948ancoms 458 . . 3 ((𝐴 ∈ FinIV𝐵𝐴) → 𝐵 ∈ V)
50 isfin4 10335 . . 3 (𝐵 ∈ V → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐵)))
5149, 50syl 17 . 2 ((𝐴 ∈ FinIV𝐵𝐴) → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐵)))
5247, 51mpbird 257 1 ((𝐴 ∈ FinIV𝐵𝐴) → 𝐵 ∈ FinIV)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  wpss 3964  c0 4339   class class class wbr 5148  cen 8981  FinIVcfin4 10318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-en 8985  df-fin4 10325
This theorem is referenced by:  domfin4  10349
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