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Theorem finsschain 9359
Description: A finite subset of the union of a superset chain is a subset of some element of the chain. A useful preliminary result for alexsub 23549 and others. (Contributed by Jeff Hankins, 25-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
finsschain (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝐵 ∈ Fin ∧ 𝐵 𝐴)) → ∃𝑧𝐴 𝐵𝑧)
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵

Proof of Theorem finsschain
Dummy variables 𝑎 𝑏 𝑐 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq1 4008 . . . . . 6 (𝑎 = ∅ → (𝑎 𝐴 ↔ ∅ ⊆ 𝐴))
2 sseq1 4008 . . . . . . 7 (𝑎 = ∅ → (𝑎𝑧 ↔ ∅ ⊆ 𝑧))
32rexbidv 3179 . . . . . 6 (𝑎 = ∅ → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 ∅ ⊆ 𝑧))
41, 3imbi12d 345 . . . . 5 (𝑎 = ∅ → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ (∅ ⊆ 𝐴 → ∃𝑧𝐴 ∅ ⊆ 𝑧)))
54imbi2d 341 . . . 4 (𝑎 = ∅ → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (∅ ⊆ 𝐴 → ∃𝑧𝐴 ∅ ⊆ 𝑧))))
6 sseq1 4008 . . . . . 6 (𝑎 = 𝑏 → (𝑎 𝐴𝑏 𝐴))
7 sseq1 4008 . . . . . . 7 (𝑎 = 𝑏 → (𝑎𝑧𝑏𝑧))
87rexbidv 3179 . . . . . 6 (𝑎 = 𝑏 → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 𝑏𝑧))
96, 8imbi12d 345 . . . . 5 (𝑎 = 𝑏 → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ (𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧)))
109imbi2d 341 . . . 4 (𝑎 = 𝑏 → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧))))
11 sseq1 4008 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 𝐴 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐴))
12 sseq1 4008 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝑧))
1312rexbidv 3179 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
1411, 13imbi12d 345 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
1514imbi2d 341 . . . 4 (𝑎 = (𝑏 ∪ {𝑐}) → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))))
16 sseq1 4008 . . . . . 6 (𝑎 = 𝐵 → (𝑎 𝐴𝐵 𝐴))
17 sseq1 4008 . . . . . . 7 (𝑎 = 𝐵 → (𝑎𝑧𝐵𝑧))
1817rexbidv 3179 . . . . . 6 (𝑎 = 𝐵 → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 𝐵𝑧))
1916, 18imbi12d 345 . . . . 5 (𝑎 = 𝐵 → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧)))
2019imbi2d 341 . . . 4 (𝑎 = 𝐵 → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧))))
21 0ss 4397 . . . . . . . 8 ∅ ⊆ 𝑧
2221rgenw 3066 . . . . . . 7 𝑧𝐴 ∅ ⊆ 𝑧
23 r19.2z 4495 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∀𝑧𝐴 ∅ ⊆ 𝑧) → ∃𝑧𝐴 ∅ ⊆ 𝑧)
2422, 23mpan2 690 . . . . . 6 (𝐴 ≠ ∅ → ∃𝑧𝐴 ∅ ⊆ 𝑧)
2524adantr 482 . . . . 5 ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → ∃𝑧𝐴 ∅ ⊆ 𝑧)
2625a1d 25 . . . 4 ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (∅ ⊆ 𝐴 → ∃𝑧𝐴 ∅ ⊆ 𝑧))
27 id 22 . . . . . . . . 9 ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → (𝑏 ∪ {𝑐}) ⊆ 𝐴)
2827unssad 4188 . . . . . . . 8 ((𝑏 ∪ {𝑐}) ⊆ 𝐴𝑏 𝐴)
2928imim1i 63 . . . . . . 7 ((𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧) → ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 𝑏𝑧))
30 sseq2 4009 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝑏𝑧𝑏𝑤))
3130cbvrexvw 3236 . . . . . . . . . 10 (∃𝑧𝐴 𝑏𝑧 ↔ ∃𝑤𝐴 𝑏𝑤)
32 simpr 486 . . . . . . . . . . . . . 14 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (𝑏 ∪ {𝑐}) ⊆ 𝐴)
3332unssbd 4189 . . . . . . . . . . . . 13 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → {𝑐} ⊆ 𝐴)
34 vex 3479 . . . . . . . . . . . . . 14 𝑐 ∈ V
3534snss 4790 . . . . . . . . . . . . 13 (𝑐 𝐴 ↔ {𝑐} ⊆ 𝐴)
3633, 35sylibr 233 . . . . . . . . . . . 12 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → 𝑐 𝐴)
37 eluni2 4913 . . . . . . . . . . . 12 (𝑐 𝐴 ↔ ∃𝑢𝐴 𝑐𝑢)
3836, 37sylib 217 . . . . . . . . . . 11 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → ∃𝑢𝐴 𝑐𝑢)
39 reeanv 3227 . . . . . . . . . . . 12 (∃𝑢𝐴𝑤𝐴 (𝑐𝑢𝑏𝑤) ↔ (∃𝑢𝐴 𝑐𝑢 ∧ ∃𝑤𝐴 𝑏𝑤))
40 simpllr 775 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → [] Or 𝐴)
41 simprlr 779 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑤𝐴)
42 simprll 778 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑢𝐴)
43 sorpssun 7720 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑤𝐴𝑢𝐴)) → (𝑤𝑢) ∈ 𝐴)
4440, 41, 42, 43syl12anc 836 . . . . . . . . . . . . . . 15 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → (𝑤𝑢) ∈ 𝐴)
45 simprrr 781 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑏𝑤)
46 simprrl 780 . . . . . . . . . . . . . . . . 17 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑐𝑢)
4746snssd 4813 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → {𝑐} ⊆ 𝑢)
48 unss12 4183 . . . . . . . . . . . . . . . 16 ((𝑏𝑤 ∧ {𝑐} ⊆ 𝑢) → (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢))
4945, 47, 48syl2anc 585 . . . . . . . . . . . . . . 15 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢))
50 sseq2 4009 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤𝑢) → ((𝑏 ∪ {𝑐}) ⊆ 𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢)))
5150rspcev 3613 . . . . . . . . . . . . . . 15 (((𝑤𝑢) ∈ 𝐴 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢)) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)
5244, 49, 51syl2anc 585 . . . . . . . . . . . . . 14 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)
5352expr 458 . . . . . . . . . . . . 13 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ (𝑢𝐴𝑤𝐴)) → ((𝑐𝑢𝑏𝑤) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5453rexlimdvva 3212 . . . . . . . . . . . 12 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (∃𝑢𝐴𝑤𝐴 (𝑐𝑢𝑏𝑤) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5539, 54biimtrrid 242 . . . . . . . . . . 11 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → ((∃𝑢𝐴 𝑐𝑢 ∧ ∃𝑤𝐴 𝑏𝑤) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5638, 55mpand 694 . . . . . . . . . 10 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (∃𝑤𝐴 𝑏𝑤 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5731, 56biimtrid 241 . . . . . . . . 9 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (∃𝑧𝐴 𝑏𝑧 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5857ex 414 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → (∃𝑧𝐴 𝑏𝑧 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
5958a2d 29 . . . . . . 7 ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 𝑏𝑧) → ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
6029, 59syl5 34 . . . . . 6 ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → ((𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧) → ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
6160a2i 14 . . . . 5 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧)) → ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
6261a1i 11 . . . 4 (𝑏 ∈ Fin → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧)) → ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))))
635, 10, 15, 20, 26, 62findcard2 9164 . . 3 (𝐵 ∈ Fin → ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧)))
6463com12 32 . 2 ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝐵 ∈ Fin → (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧)))
6564imp32 420 1 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝐵 ∈ Fin ∧ 𝐵 𝐴)) → ∃𝑧𝐴 𝐵𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  cun 3947  wss 3949  c0 4323  {csn 4629   cuni 4909   Or wor 5588   [] crpss 7712  Fincfn 8939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-rpss 7713  df-om 7856  df-en 8940  df-fin 8943
This theorem is referenced by:  alexsubALTlem2  23552
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