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Theorem ssfin4 9725
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 766 . . . 4 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝐴 ∈ FinIV)
2 pssss 4026 . . . . . . . . 9 (𝑥𝐵𝑥𝐵)
3 simpr 488 . . . . . . . . 9 ((𝐴 ∈ FinIV𝐵𝐴) → 𝐵𝐴)
42, 3sylan9ssr 3932 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → 𝑥𝐴)
5 difssd 4063 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → (𝐴𝐵) ⊆ 𝐴)
64, 5unssd 4116 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → (𝑥 ∪ (𝐴𝐵)) ⊆ 𝐴)
7 pssnel 4381 . . . . . . . . 9 (𝑥𝐵 → ∃𝑐(𝑐𝐵 ∧ ¬ 𝑐𝑥))
87adantl 485 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → ∃𝑐(𝑐𝐵 ∧ ¬ 𝑐𝑥))
9 simpllr 775 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → 𝐵𝐴)
10 simprl 770 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → 𝑐𝐵)
119, 10sseldd 3919 . . . . . . . . . 10 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → 𝑐𝐴)
12 simprr 772 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝑐𝑥)
13 elndif 4059 . . . . . . . . . . . 12 (𝑐𝐵 → ¬ 𝑐 ∈ (𝐴𝐵))
1413ad2antrl 727 . . . . . . . . . . 11 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝑐 ∈ (𝐴𝐵))
15 ioran 981 . . . . . . . . . . . 12 (¬ (𝑐𝑥𝑐 ∈ (𝐴𝐵)) ↔ (¬ 𝑐𝑥 ∧ ¬ 𝑐 ∈ (𝐴𝐵)))
16 elun 4079 . . . . . . . . . . . 12 (𝑐 ∈ (𝑥 ∪ (𝐴𝐵)) ↔ (𝑐𝑥𝑐 ∈ (𝐴𝐵)))
1715, 16xchnxbir 336 . . . . . . . . . . 11 𝑐 ∈ (𝑥 ∪ (𝐴𝐵)) ↔ (¬ 𝑐𝑥 ∧ ¬ 𝑐 ∈ (𝐴𝐵)))
1812, 14, 17sylanbrc 586 . . . . . . . . . 10 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝑐 ∈ (𝑥 ∪ (𝐴𝐵)))
19 nelneq2 2918 . . . . . . . . . 10 ((𝑐𝐴 ∧ ¬ 𝑐 ∈ (𝑥 ∪ (𝐴𝐵))) → ¬ 𝐴 = (𝑥 ∪ (𝐴𝐵)))
2011, 18, 19syl2anc 587 . . . . . . . . 9 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ 𝐴 = (𝑥 ∪ (𝐴𝐵)))
21 eqcom 2808 . . . . . . . . 9 (𝐴 = (𝑥 ∪ (𝐴𝐵)) ↔ (𝑥 ∪ (𝐴𝐵)) = 𝐴)
2220, 21sylnib 331 . . . . . . . 8 ((((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) ∧ (𝑐𝐵 ∧ ¬ 𝑐𝑥)) → ¬ (𝑥 ∪ (𝐴𝐵)) = 𝐴)
238, 22exlimddv 1936 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → ¬ (𝑥 ∪ (𝐴𝐵)) = 𝐴)
24 dfpss2 4016 . . . . . . 7 ((𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴 ↔ ((𝑥 ∪ (𝐴𝐵)) ⊆ 𝐴 ∧ ¬ (𝑥 ∪ (𝐴𝐵)) = 𝐴))
256, 23, 24sylanbrc 586 . . . . . 6 (((𝐴 ∈ FinIV𝐵𝐴) ∧ 𝑥𝐵) → (𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴)
2625adantrr 716 . . . . 5 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴)
27 simprr 772 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝑥𝐵)
28 difexg 5198 . . . . . . . 8 (𝐴 ∈ FinIV → (𝐴𝐵) ∈ V)
29 enrefg 8528 . . . . . . . 8 ((𝐴𝐵) ∈ V → (𝐴𝐵) ≈ (𝐴𝐵))
301, 28, 293syl 18 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝐴𝐵) ≈ (𝐴𝐵))
312ad2antrl 727 . . . . . . . . . 10 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝑥𝐵)
32 ssinss1 4167 . . . . . . . . . 10 (𝑥𝐵 → (𝑥𝐴) ⊆ 𝐵)
3331, 32syl 17 . . . . . . . . 9 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥𝐴) ⊆ 𝐵)
34 inssdif0 4286 . . . . . . . . 9 ((𝑥𝐴) ⊆ 𝐵 ↔ (𝑥 ∩ (𝐴𝐵)) = ∅)
3533, 34sylib 221 . . . . . . . 8 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∩ (𝐴𝐵)) = ∅)
36 disjdif 4382 . . . . . . . 8 (𝐵 ∩ (𝐴𝐵)) = ∅
3735, 36jctir 524 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → ((𝑥 ∩ (𝐴𝐵)) = ∅ ∧ (𝐵 ∩ (𝐴𝐵)) = ∅))
38 unen 8583 . . . . . . 7 (((𝑥𝐵 ∧ (𝐴𝐵) ≈ (𝐴𝐵)) ∧ ((𝑥 ∩ (𝐴𝐵)) = ∅ ∧ (𝐵 ∩ (𝐴𝐵)) = ∅)) → (𝑥 ∪ (𝐴𝐵)) ≈ (𝐵 ∪ (𝐴𝐵)))
3927, 30, 37, 38syl21anc 836 . . . . . 6 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∪ (𝐴𝐵)) ≈ (𝐵 ∪ (𝐴𝐵)))
40 simplr 768 . . . . . . 7 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → 𝐵𝐴)
41 undif 4391 . . . . . . 7 (𝐵𝐴 ↔ (𝐵 ∪ (𝐴𝐵)) = 𝐴)
4240, 41sylib 221 . . . . . 6 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝐵 ∪ (𝐴𝐵)) = 𝐴)
4339, 42breqtrd 5059 . . . . 5 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → (𝑥 ∪ (𝐴𝐵)) ≈ 𝐴)
44 fin4i 9713 . . . . 5 (((𝑥 ∪ (𝐴𝐵)) ⊊ 𝐴 ∧ (𝑥 ∪ (𝐴𝐵)) ≈ 𝐴) → ¬ 𝐴 ∈ FinIV)
4526, 43, 44syl2anc 587 . . . 4 (((𝐴 ∈ FinIV𝐵𝐴) ∧ (𝑥𝐵𝑥𝐵)) → ¬ 𝐴 ∈ FinIV)
461, 45pm2.65da 816 . . 3 ((𝐴 ∈ FinIV𝐵𝐴) → ¬ (𝑥𝐵𝑥𝐵))
4746nexdv 1937 . 2 ((𝐴 ∈ FinIV𝐵𝐴) → ¬ ∃𝑥(𝑥𝐵𝑥𝐵))
48 ssexg 5194 . . . 4 ((𝐵𝐴𝐴 ∈ FinIV) → 𝐵 ∈ V)
4948ancoms 462 . . 3 ((𝐴 ∈ FinIV𝐵𝐴) → 𝐵 ∈ V)
50 isfin4 9712 . . 3 (𝐵 ∈ V → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐵)))
5149, 50syl 17 . 2 ((𝐴 ∈ FinIV𝐵𝐴) → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥𝐵𝑥𝐵)))
5247, 51mpbird 260 1 ((𝐴 ∈ FinIV𝐵𝐴) → 𝐵 ∈ FinIV)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wex 1781  wcel 2112  Vcvv 3444  cdif 3881  cun 3882  cin 3883  wss 3884  wpss 3885  c0 4246   class class class wbr 5033  cen 8493  FinIVcfin4 9695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-en 8497  df-fin4 9702
This theorem is referenced by:  domfin4  9726
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