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Theorem finsschain 9358
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 23548 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 4007 . . . . . 6 (𝑎 = ∅ → (𝑎 𝐴 ↔ ∅ ⊆ 𝐴))
2 sseq1 4007 . . . . . . 7 (𝑎 = ∅ → (𝑎𝑧 ↔ ∅ ⊆ 𝑧))
32rexbidv 3178 . . . . . 6 (𝑎 = ∅ → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 ∅ ⊆ 𝑧))
41, 3imbi12d 344 . . . . 5 (𝑎 = ∅ → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ (∅ ⊆ 𝐴 → ∃𝑧𝐴 ∅ ⊆ 𝑧)))
54imbi2d 340 . . . 4 (𝑎 = ∅ → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (∅ ⊆ 𝐴 → ∃𝑧𝐴 ∅ ⊆ 𝑧))))
6 sseq1 4007 . . . . . 6 (𝑎 = 𝑏 → (𝑎 𝐴𝑏 𝐴))
7 sseq1 4007 . . . . . . 7 (𝑎 = 𝑏 → (𝑎𝑧𝑏𝑧))
87rexbidv 3178 . . . . . 6 (𝑎 = 𝑏 → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 𝑏𝑧))
96, 8imbi12d 344 . . . . 5 (𝑎 = 𝑏 → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ (𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧)))
109imbi2d 340 . . . 4 (𝑎 = 𝑏 → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧))))
11 sseq1 4007 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 𝐴 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐴))
12 sseq1 4007 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝑧))
1312rexbidv 3178 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
1411, 13imbi12d 344 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
1514imbi2d 340 . . . 4 (𝑎 = (𝑏 ∪ {𝑐}) → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))))
16 sseq1 4007 . . . . . 6 (𝑎 = 𝐵 → (𝑎 𝐴𝐵 𝐴))
17 sseq1 4007 . . . . . . 7 (𝑎 = 𝐵 → (𝑎𝑧𝐵𝑧))
1817rexbidv 3178 . . . . . 6 (𝑎 = 𝐵 → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 𝐵𝑧))
1916, 18imbi12d 344 . . . . 5 (𝑎 = 𝐵 → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧)))
2019imbi2d 340 . . . 4 (𝑎 = 𝐵 → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧))))
21 0ss 4396 . . . . . . . 8 ∅ ⊆ 𝑧
2221rgenw 3065 . . . . . . 7 𝑧𝐴 ∅ ⊆ 𝑧
23 r19.2z 4494 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∀𝑧𝐴 ∅ ⊆ 𝑧) → ∃𝑧𝐴 ∅ ⊆ 𝑧)
2422, 23mpan2 689 . . . . . 6 (𝐴 ≠ ∅ → ∃𝑧𝐴 ∅ ⊆ 𝑧)
2524adantr 481 . . . . 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 4187 . . . . . . . 8 ((𝑏 ∪ {𝑐}) ⊆ 𝐴𝑏 𝐴)
2928imim1i 63 . . . . . . 7 ((𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧) → ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 𝑏𝑧))
30 sseq2 4008 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝑏𝑧𝑏𝑤))
3130cbvrexvw 3235 . . . . . . . . . 10 (∃𝑧𝐴 𝑏𝑧 ↔ ∃𝑤𝐴 𝑏𝑤)
32 simpr 485 . . . . . . . . . . . . . 14 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (𝑏 ∪ {𝑐}) ⊆ 𝐴)
3332unssbd 4188 . . . . . . . . . . . . 13 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → {𝑐} ⊆ 𝐴)
34 vex 3478 . . . . . . . . . . . . . 14 𝑐 ∈ V
3534snss 4789 . . . . . . . . . . . . 13 (𝑐 𝐴 ↔ {𝑐} ⊆ 𝐴)
3633, 35sylibr 233 . . . . . . . . . . . 12 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → 𝑐 𝐴)
37 eluni2 4912 . . . . . . . . . . . 12 (𝑐 𝐴 ↔ ∃𝑢𝐴 𝑐𝑢)
3836, 37sylib 217 . . . . . . . . . . 11 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → ∃𝑢𝐴 𝑐𝑢)
39 reeanv 3226 . . . . . . . . . . . 12 (∃𝑢𝐴𝑤𝐴 (𝑐𝑢𝑏𝑤) ↔ (∃𝑢𝐴 𝑐𝑢 ∧ ∃𝑤𝐴 𝑏𝑤))
40 simpllr 774 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → [] Or 𝐴)
41 simprlr 778 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑤𝐴)
42 simprll 777 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑢𝐴)
43 sorpssun 7719 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑤𝐴𝑢𝐴)) → (𝑤𝑢) ∈ 𝐴)
4440, 41, 42, 43syl12anc 835 . . . . . . . . . . . . . . 15 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → (𝑤𝑢) ∈ 𝐴)
45 simprrr 780 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑏𝑤)
46 simprrl 779 . . . . . . . . . . . . . . . . 17 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑐𝑢)
4746snssd 4812 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → {𝑐} ⊆ 𝑢)
48 unss12 4182 . . . . . . . . . . . . . . . 16 ((𝑏𝑤 ∧ {𝑐} ⊆ 𝑢) → (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢))
4945, 47, 48syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢))
50 sseq2 4008 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤𝑢) → ((𝑏 ∪ {𝑐}) ⊆ 𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢)))
5150rspcev 3612 . . . . . . . . . . . . . . 15 (((𝑤𝑢) ∈ 𝐴 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢)) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)
5244, 49, 51syl2anc 584 . . . . . . . . . . . . . 14 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)
5352expr 457 . . . . . . . . . . . . 13 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ (𝑢𝐴𝑤𝐴)) → ((𝑐𝑢𝑏𝑤) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5453rexlimdvva 3211 . . . . . . . . . . . 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 693 . . . . . . . . . 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 413 . . . . . . . 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 9163 . . 3 (𝐵 ∈ Fin → ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧)))
6463com12 32 . 2 ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝐵 ∈ Fin → (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧)))
6564imp32 419 1 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝐵 ∈ Fin ∧ 𝐵 𝐴)) → ∃𝑧𝐴 𝐵𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2940  wral 3061  wrex 3070  cun 3946  wss 3948  c0 4322  {csn 4628   cuni 4908   Or wor 5587   [] crpss 7711  Fincfn 8938
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-rpss 7712  df-om 7855  df-en 8939  df-fin 8942
This theorem is referenced by:  alexsubALTlem2  23551
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