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Theorem finsschain 8224
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 21768 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 3610 . . . . . 6 (𝑎 = ∅ → (𝑎 𝐴 ↔ ∅ ⊆ 𝐴))
2 sseq1 3610 . . . . . . 7 (𝑎 = ∅ → (𝑎𝑧 ↔ ∅ ⊆ 𝑧))
32rexbidv 3046 . . . . . 6 (𝑎 = ∅ → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 ∅ ⊆ 𝑧))
41, 3imbi12d 334 . . . . 5 (𝑎 = ∅ → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ (∅ ⊆ 𝐴 → ∃𝑧𝐴 ∅ ⊆ 𝑧)))
54imbi2d 330 . . . 4 (𝑎 = ∅ → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (∅ ⊆ 𝐴 → ∃𝑧𝐴 ∅ ⊆ 𝑧))))
6 sseq1 3610 . . . . . 6 (𝑎 = 𝑏 → (𝑎 𝐴𝑏 𝐴))
7 sseq1 3610 . . . . . . 7 (𝑎 = 𝑏 → (𝑎𝑧𝑏𝑧))
87rexbidv 3046 . . . . . 6 (𝑎 = 𝑏 → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 𝑏𝑧))
96, 8imbi12d 334 . . . . 5 (𝑎 = 𝑏 → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ (𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧)))
109imbi2d 330 . . . 4 (𝑎 = 𝑏 → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧))))
11 sseq1 3610 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 𝐴 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐴))
12 sseq1 3610 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝑧))
1312rexbidv 3046 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
1411, 13imbi12d 334 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
1514imbi2d 330 . . . 4 (𝑎 = (𝑏 ∪ {𝑐}) → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))))
16 sseq1 3610 . . . . . 6 (𝑎 = 𝐵 → (𝑎 𝐴𝐵 𝐴))
17 sseq1 3610 . . . . . . 7 (𝑎 = 𝐵 → (𝑎𝑧𝐵𝑧))
1817rexbidv 3046 . . . . . 6 (𝑎 = 𝐵 → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 𝐵𝑧))
1916, 18imbi12d 334 . . . . 5 (𝑎 = 𝐵 → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧)))
2019imbi2d 330 . . . 4 (𝑎 = 𝐵 → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧))))
21 0ss 3949 . . . . . . . 8 ∅ ⊆ 𝑧
2221rgenw 2919 . . . . . . 7 𝑧𝐴 ∅ ⊆ 𝑧
23 r19.2z 4037 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∀𝑧𝐴 ∅ ⊆ 𝑧) → ∃𝑧𝐴 ∅ ⊆ 𝑧)
2422, 23mpan2 706 . . . . . 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 3773 . . . . . . . 8 ((𝑏 ∪ {𝑐}) ⊆ 𝐴𝑏 𝐴)
2928imim1i 63 . . . . . . 7 ((𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧) → ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 𝑏𝑧))
30 sseq2 3611 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝑏𝑧𝑏𝑤))
3130cbvrexv 3163 . . . . . . . . . 10 (∃𝑧𝐴 𝑏𝑧 ↔ ∃𝑤𝐴 𝑏𝑤)
32 simpr 477 . . . . . . . . . . . . . 14 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (𝑏 ∪ {𝑐}) ⊆ 𝐴)
3332unssbd 3774 . . . . . . . . . . . . 13 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → {𝑐} ⊆ 𝐴)
34 vex 3192 . . . . . . . . . . . . . 14 𝑐 ∈ V
3534snss 4291 . . . . . . . . . . . . 13 (𝑐 𝐴 ↔ {𝑐} ⊆ 𝐴)
3633, 35sylibr 224 . . . . . . . . . . . 12 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → 𝑐 𝐴)
37 eluni2 4411 . . . . . . . . . . . 12 (𝑐 𝐴 ↔ ∃𝑢𝐴 𝑐𝑢)
3836, 37sylib 208 . . . . . . . . . . 11 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → ∃𝑢𝐴 𝑐𝑢)
39 reeanv 3100 . . . . . . . . . . . 12 (∃𝑢𝐴𝑤𝐴 (𝑐𝑢𝑏𝑤) ↔ (∃𝑢𝐴 𝑐𝑢 ∧ ∃𝑤𝐴 𝑏𝑤))
40 simpllr 798 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → [] Or 𝐴)
41 simprlr 802 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑤𝐴)
42 simprll 801 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑢𝐴)
43 sorpssun 6904 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑤𝐴𝑢𝐴)) → (𝑤𝑢) ∈ 𝐴)
4440, 41, 42, 43syl12anc 1321 . . . . . . . . . . . . . . 15 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → (𝑤𝑢) ∈ 𝐴)
45 simprrr 804 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑏𝑤)
46 simprrl 803 . . . . . . . . . . . . . . . . 17 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑐𝑢)
4746snssd 4314 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → {𝑐} ⊆ 𝑢)
48 unss12 3768 . . . . . . . . . . . . . . . 16 ((𝑏𝑤 ∧ {𝑐} ⊆ 𝑢) → (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢))
4945, 47, 48syl2anc 692 . . . . . . . . . . . . . . 15 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢))
50 sseq2 3611 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤𝑢) → ((𝑏 ∪ {𝑐}) ⊆ 𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢)))
5150rspcev 3298 . . . . . . . . . . . . . . 15 (((𝑤𝑢) ∈ 𝐴 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢)) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)
5244, 49, 51syl2anc 692 . . . . . . . . . . . . . 14 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)
5352expr 642 . . . . . . . . . . . . 13 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ (𝑢𝐴𝑤𝐴)) → ((𝑐𝑢𝑏𝑤) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5453rexlimdvva 3032 . . . . . . . . . . . 12 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (∃𝑢𝐴𝑤𝐴 (𝑐𝑢𝑏𝑤) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5539, 54syl5bir 233 . . . . . . . . . . 11 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → ((∃𝑢𝐴 𝑐𝑢 ∧ ∃𝑤𝐴 𝑏𝑤) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5638, 55mpand 710 . . . . . . . . . 10 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (∃𝑤𝐴 𝑏𝑤 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5731, 56syl5bi 232 . . . . . . . . 9 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (∃𝑧𝐴 𝑏𝑧 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5857ex 450 . . . . . . . 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 8151 . . 3 (𝐵 ∈ Fin → ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧)))
6463com12 32 . 2 ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝐵 ∈ Fin → (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧)))
6564imp32 449 1 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝐵 ∈ Fin ∧ 𝐵 𝐴)) → ∃𝑧𝐴 𝐵𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  cun 3557  wss 3559  c0 3896  {csn 4153   cuni 4407   Or wor 4999   [] crpss 6896  Fincfn 7906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-rpss 6897  df-om 7020  df-1o 7512  df-er 7694  df-en 7907  df-fin 7910
This theorem is referenced by:  alexsubALTlem2  21771
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