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Theorem finsschain 8156
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 21659 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 3589 . . . . . 6 (𝑎 = ∅ → (𝑎 𝐴 ↔ ∅ ⊆ 𝐴))
2 sseq1 3589 . . . . . . 7 (𝑎 = ∅ → (𝑎𝑧 ↔ ∅ ⊆ 𝑧))
32rexbidv 3034 . . . . . 6 (𝑎 = ∅ → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 ∅ ⊆ 𝑧))
41, 3imbi12d 333 . . . . 5 (𝑎 = ∅ → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ (∅ ⊆ 𝐴 → ∃𝑧𝐴 ∅ ⊆ 𝑧)))
54imbi2d 329 . . . 4 (𝑎 = ∅ → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (∅ ⊆ 𝐴 → ∃𝑧𝐴 ∅ ⊆ 𝑧))))
6 sseq1 3589 . . . . . 6 (𝑎 = 𝑏 → (𝑎 𝐴𝑏 𝐴))
7 sseq1 3589 . . . . . . 7 (𝑎 = 𝑏 → (𝑎𝑧𝑏𝑧))
87rexbidv 3034 . . . . . 6 (𝑎 = 𝑏 → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 𝑏𝑧))
96, 8imbi12d 333 . . . . 5 (𝑎 = 𝑏 → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ (𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧)))
109imbi2d 329 . . . 4 (𝑎 = 𝑏 → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧))))
11 sseq1 3589 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 𝐴 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐴))
12 sseq1 3589 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝑧))
1312rexbidv 3034 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
1411, 13imbi12d 333 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
1514imbi2d 329 . . . 4 (𝑎 = (𝑏 ∪ {𝑐}) → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))))
16 sseq1 3589 . . . . . 6 (𝑎 = 𝐵 → (𝑎 𝐴𝐵 𝐴))
17 sseq1 3589 . . . . . . 7 (𝑎 = 𝐵 → (𝑎𝑧𝐵𝑧))
1817rexbidv 3034 . . . . . 6 (𝑎 = 𝐵 → (∃𝑧𝐴 𝑎𝑧 ↔ ∃𝑧𝐴 𝐵𝑧))
1916, 18imbi12d 333 . . . . 5 (𝑎 = 𝐵 → ((𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧) ↔ (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧)))
2019imbi2d 329 . . . 4 (𝑎 = 𝐵 → (((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝑎 𝐴 → ∃𝑧𝐴 𝑎𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧))))
21 0ss 3924 . . . . . . . 8 ∅ ⊆ 𝑧
2221rgenw 2908 . . . . . . 7 𝑧𝐴 ∅ ⊆ 𝑧
23 r19.2z 4012 . . . . . . 7 ((𝐴 ≠ ∅ ∧ ∀𝑧𝐴 ∅ ⊆ 𝑧) → ∃𝑧𝐴 ∅ ⊆ 𝑧)
2422, 23mpan2 703 . . . . . 6 (𝐴 ≠ ∅ → ∃𝑧𝐴 ∅ ⊆ 𝑧)
2524adantr 480 . . . . 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 3752 . . . . . . . 8 ((𝑏 ∪ {𝑐}) ⊆ 𝐴𝑏 𝐴)
2928imim1i 61 . . . . . . 7 ((𝑏 𝐴 → ∃𝑧𝐴 𝑏𝑧) → ((𝑏 ∪ {𝑐}) ⊆ 𝐴 → ∃𝑧𝐴 𝑏𝑧))
30 sseq2 3590 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝑏𝑧𝑏𝑤))
3130cbvrexv 3148 . . . . . . . . . 10 (∃𝑧𝐴 𝑏𝑧 ↔ ∃𝑤𝐴 𝑏𝑤)
32 simpr 476 . . . . . . . . . . . . . 14 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (𝑏 ∪ {𝑐}) ⊆ 𝐴)
3332unssbd 3753 . . . . . . . . . . . . 13 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → {𝑐} ⊆ 𝐴)
34 vex 3176 . . . . . . . . . . . . . 14 𝑐 ∈ V
3534snss 4259 . . . . . . . . . . . . 13 (𝑐 𝐴 ↔ {𝑐} ⊆ 𝐴)
3633, 35sylibr 223 . . . . . . . . . . . 12 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → 𝑐 𝐴)
37 eluni2 4376 . . . . . . . . . . . 12 (𝑐 𝐴 ↔ ∃𝑢𝐴 𝑐𝑢)
3836, 37sylib 207 . . . . . . . . . . 11 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → ∃𝑢𝐴 𝑐𝑢)
39 reeanv 3086 . . . . . . . . . . . 12 (∃𝑢𝐴𝑤𝐴 (𝑐𝑢𝑏𝑤) ↔ (∃𝑢𝐴 𝑐𝑢 ∧ ∃𝑤𝐴 𝑏𝑤))
40 simpllr 795 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → [] Or 𝐴)
41 simprlr 799 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑤𝐴)
42 simprll 798 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑢𝐴)
43 sorpssun 6842 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑤𝐴𝑢𝐴)) → (𝑤𝑢) ∈ 𝐴)
4440, 41, 42, 43syl12anc 1316 . . . . . . . . . . . . . . 15 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → (𝑤𝑢) ∈ 𝐴)
45 simprrr 801 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑏𝑤)
46 simprrl 800 . . . . . . . . . . . . . . . . 17 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → 𝑐𝑢)
4746snssd 4281 . . . . . . . . . . . . . . . 16 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → {𝑐} ⊆ 𝑢)
48 unss12 3747 . . . . . . . . . . . . . . . 16 ((𝑏𝑤 ∧ {𝑐} ⊆ 𝑢) → (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢))
4945, 47, 48syl2anc 691 . . . . . . . . . . . . . . 15 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢))
50 sseq2 3590 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑤𝑢) → ((𝑏 ∪ {𝑐}) ⊆ 𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢)))
5150rspcev 3282 . . . . . . . . . . . . . . 15 (((𝑤𝑢) ∈ 𝐴 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑤𝑢)) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)
5244, 49, 51syl2anc 691 . . . . . . . . . . . . . 14 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ ((𝑢𝐴𝑤𝐴) ∧ (𝑐𝑢𝑏𝑤))) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)
5352expr 641 . . . . . . . . . . . . 13 ((((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) ∧ (𝑢𝐴𝑤𝐴)) → ((𝑐𝑢𝑏𝑤) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5453rexlimdvva 3020 . . . . . . . . . . . 12 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (∃𝑢𝐴𝑤𝐴 (𝑐𝑢𝑏𝑤) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5539, 54syl5bir 232 . . . . . . . . . . 11 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → ((∃𝑢𝐴 𝑐𝑢 ∧ ∃𝑤𝐴 𝑏𝑤) → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5638, 55mpand 707 . . . . . . . . . 10 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (∃𝑤𝐴 𝑏𝑤 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5731, 56syl5bi 231 . . . . . . . . 9 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐴) → (∃𝑧𝐴 𝑏𝑧 → ∃𝑧𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
5857ex 449 . . . . . . . 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 33 . . . . . 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 8085 . . 3 (𝐵 ∈ Fin → ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧)))
6463com12 32 . 2 ((𝐴 ≠ ∅ ∧ [] Or 𝐴) → (𝐵 ∈ Fin → (𝐵 𝐴 → ∃𝑧𝐴 𝐵𝑧)))
6564imp32 448 1 (((𝐴 ≠ ∅ ∧ [] Or 𝐴) ∧ (𝐵 ∈ Fin ∧ 𝐵 𝐴)) → ∃𝑧𝐴 𝐵𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wne 2780  wral 2896  wrex 2897  cun 3538  wss 3540  c0 3874  {csn 4125   cuni 4372   Or wor 4958   [] crpss 6834  Fincfn 7841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-rpss 6835  df-om 6958  df-1o 7447  df-er 7629  df-en 7842  df-fin 7845
This theorem is referenced by:  alexsubALTlem2  21662
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