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Theorem finsschain 8561
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 22257 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 3845 . . . . . 6 (𝑎 = ∅ → (𝑎 𝐴 ↔ ∅ ⊆ 𝐴))
2 sseq1 3845 . . . . . . 7 (𝑎 = ∅ → (𝑎𝑧 ↔ ∅ ⊆ 𝑧))
32rexbidv 3237 . . . . . 6 (𝑎 = ∅ → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 ∅ ⊆ 𝑧))
41, 3imbi12d 336 . . . . 5 (𝑎 = ∅ → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ (∅ ⊆ 𝐴 → ∃𝑧𝐴 ∅ ⊆ 𝑧)))
54imbi2d 332 . . . 4 (𝑎 = ∅ → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (∅ ⊆ 𝐴 → ∃𝑧𝐴 ∅ ⊆ 𝑧))))
6 sseq1 3845 . . . . . 6 (𝑎 = 𝑏 → (𝑎 𝐴𝑏 𝐴))
7 sseq1 3845 . . . . . . 7 (𝑎 = 𝑏 → (𝑎𝑧𝑏𝑧))
87rexbidv 3237 . . . . . 6 (𝑎 = 𝑏 → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 𝑏𝑧))
96, 8imbi12d 336 . . . . 5 (𝑎 = 𝑏 → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ (𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧)))
109imbi2d 332 . . . 4 (𝑎 = 𝑏 → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧))))
11 sseq1 3845 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 𝐴 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐴))
12 sseq1 3845 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝑧))
1312rexbidv 3237 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
1411, 13imbi12d 336 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
1514imbi2d 332 . . . 4 (𝑎 = (𝑏 ∪ {𝑐}) → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))))
16 sseq1 3845 . . . . . 6 (𝑎 = 𝐵 → (𝑎 𝐴𝐵 𝐴))
17 sseq1 3845 . . . . . . 7 (𝑎 = 𝐵 → (𝑎𝑧𝐵𝑧))
1817rexbidv 3237 . . . . . 6 (𝑎 = 𝐵 → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 𝐵𝑧))
1916, 18imbi12d 336 . . . . 5 (𝑎 = 𝐵 → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧)))
2019imbi2d 332 . . . 4 (𝑎 = 𝐵 → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧))))
21 0ss 4198 . . . . . . . 8 ∅ ⊆ 𝑧
2221rgenw 3106 . . . . . . 7 𝑧𝐴 ∅ ⊆ 𝑧
23 r19.2z 4283 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∀𝑧𝐴 ∅ ⊆ 𝑧) → ∃𝑧𝐴 ∅ ⊆ 𝑧)
2422, 23mpan2 681 . . . . . 6 (𝐴 ≠ ∅ → ∃𝑧𝐴 ∅ ⊆ 𝑧)
2524adantr 474 . . . . 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 4013 . . . . . . . 8 ((𝑏 ∪ {𝑐}) ⊆ 𝐴𝑏 𝐴)
2928imim1i 63 . . . . . . 7 ((𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧) → ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 𝑏𝑧))
30 sseq2 3846 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝑏𝑧𝑏𝑤))
3130cbvrexv 3368 . . . . . . . . . 10 (∃𝑧𝐴 𝑏𝑧 ↔ ∃𝑤𝐴 𝑏𝑤)
32 simpr 479 . . . . . . . . . . . . . 14 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (𝑏 ∪ {𝑐}) ⊆ 𝐴)
3332unssbd 4014 . . . . . . . . . . . . 13 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → {𝑐} ⊆ 𝐴)
34 vex 3401 . . . . . . . . . . . . . 14 𝑐 ∈ V
3534snss 4549 . . . . . . . . . . . . 13 (𝑐 𝐴 ↔ {𝑐} ⊆ 𝐴)
3633, 35sylibr 226 . . . . . . . . . . . 12 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → 𝑐 𝐴)
37 eluni2 4675 . . . . . . . . . . . 12 (𝑐 𝐴 ↔ ∃𝑢𝐴 𝑐𝑢)
3836, 37sylib 210 . . . . . . . . . . 11 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → ∃𝑢𝐴 𝑐𝑢)
39 reeanv 3293 . . . . . . . . . . . 12 (∃𝑢𝐴𝑤𝐴 (𝑐𝑢𝑏𝑤) ↔ (∃𝑢𝐴 𝑐𝑢 ∧ ∃𝑤𝐴 𝑏𝑤))
40 simpllr 766 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → [] Or 𝐴)
41 simprlr 770 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑤𝐴)
42 simprll 769 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑢𝐴)
43 sorpssun 7221 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑤𝐴𝑢𝐴)) → (𝑤𝑢) ∈ 𝐴)
4440, 41, 42, 43syl12anc 827 . . . . . . . . . . . . . . 15 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → (𝑤𝑢) ∈ 𝐴)
45 simprrr 772 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑏𝑤)
46 simprrl 771 . . . . . . . . . . . . . . . . 17 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑐𝑢)
4746snssd 4571 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → {𝑐} ⊆ 𝑢)
48 unss12 4008 . . . . . . . . . . . . . . . 16 ((𝑏𝑤 ∧ {𝑐} ⊆ 𝑢) → (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢))
4945, 47, 48syl2anc 579 . . . . . . . . . . . . . . 15 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢))
50 sseq2 3846 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤𝑢) → ((𝑏 ∪ {𝑐}) ⊆ 𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢)))
5150rspcev 3511 . . . . . . . . . . . . . . 15 (((𝑤𝑢) ∈ 𝐴 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢)) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)
5244, 49, 51syl2anc 579 . . . . . . . . . . . . . 14 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)
5352expr 450 . . . . . . . . . . . . 13 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ (𝑢𝐴𝑤𝐴)) → ((𝑐𝑢𝑏𝑤) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5453rexlimdvva 3221 . . . . . . . . . . . 12 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (∃𝑢𝐴𝑤𝐴 (𝑐𝑢𝑏𝑤) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5539, 54syl5bir 235 . . . . . . . . . . 11 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → ((∃𝑢𝐴 𝑐𝑢 ∧ ∃𝑤𝐴 𝑏𝑤) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5638, 55mpand 685 . . . . . . . . . 10 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (∃𝑤𝐴 𝑏𝑤 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5731, 56syl5bi 234 . . . . . . . . 9 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (∃𝑧𝐴 𝑏𝑧 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5857ex 403 . . . . . . . 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 8488 . . 3 (𝐵 ∈ Fin → ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧)))
6463com12 32 . 2 ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝐵 ∈ Fin → (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧)))
6564imp32 411 1 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝐵 ∈ Fin ∧ 𝐵 𝐴)) → ∃𝑧𝐴 𝐵𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  wne 2969  wral 3090  wrex 3091  cun 3790  wss 3792  c0 4141  {csn 4398   cuni 4671   Or wor 5273   [] crpss 7213  Fincfn 8241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-rpss 7214  df-om 7344  df-1o 7843  df-er 8026  df-en 8242  df-fin 8245
This theorem is referenced by:  alexsubALTlem2  22260
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