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Theorem sorpssuni 7751
Description: In a chain of sets, a maximal element is the union of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
sorpssuni ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣 𝑌𝑌))
Distinct variable group:   𝑢,𝑌,𝑣

Proof of Theorem sorpssuni
StepHypRef Expression
1 sorpssi 7748 . . . . . . . . . 10 (( [] Or 𝑌 ∧ (𝑢𝑌𝑣𝑌)) → (𝑢𝑣𝑣𝑢))
21anassrs 467 . . . . . . . . 9 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (𝑢𝑣𝑣𝑢))
3 sspss 4112 . . . . . . . . . . 11 (𝑢𝑣 ↔ (𝑢𝑣𝑢 = 𝑣))
4 orel1 888 . . . . . . . . . . . 12 𝑢𝑣 → ((𝑢𝑣𝑢 = 𝑣) → 𝑢 = 𝑣))
5 eqimss2 4055 . . . . . . . . . . . 12 (𝑢 = 𝑣𝑣𝑢)
64, 5syl6com 37 . . . . . . . . . . 11 ((𝑢𝑣𝑢 = 𝑣) → (¬ 𝑢𝑣𝑣𝑢))
73, 6sylbi 217 . . . . . . . . . 10 (𝑢𝑣 → (¬ 𝑢𝑣𝑣𝑢))
8 ax-1 6 . . . . . . . . . 10 (𝑣𝑢 → (¬ 𝑢𝑣𝑣𝑢))
97, 8jaoi 857 . . . . . . . . 9 ((𝑢𝑣𝑣𝑢) → (¬ 𝑢𝑣𝑣𝑢))
102, 9syl 17 . . . . . . . 8 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (¬ 𝑢𝑣𝑣𝑢))
1110ralimdva 3165 . . . . . . 7 (( [] Or 𝑌𝑢𝑌) → (∀𝑣𝑌 ¬ 𝑢𝑣 → ∀𝑣𝑌 𝑣𝑢))
12113impia 1116 . . . . . 6 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → ∀𝑣𝑌 𝑣𝑢)
13 unissb 4944 . . . . . 6 ( 𝑌𝑢 ↔ ∀𝑣𝑌 𝑣𝑢)
1412, 13sylibr 234 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑌𝑢)
15 elssuni 4942 . . . . . 6 (𝑢𝑌𝑢 𝑌)
16153ad2ant2 1133 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑢 𝑌)
1714, 16eqssd 4013 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑌 = 𝑢)
18 simp2 1136 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑢𝑌)
1917, 18eqeltrd 2839 . . 3 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑌𝑌)
2019rexlimdv3a 3157 . 2 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣 𝑌𝑌))
21 elssuni 4942 . . . . 5 (𝑣𝑌𝑣 𝑌)
22 ssnpss 4116 . . . . 5 (𝑣 𝑌 → ¬ 𝑌𝑣)
2321, 22syl 17 . . . 4 (𝑣𝑌 → ¬ 𝑌𝑣)
2423rgen 3061 . . 3 𝑣𝑌 ¬ 𝑌𝑣
25 psseq1 4100 . . . . . 6 (𝑢 = 𝑌 → (𝑢𝑣 𝑌𝑣))
2625notbid 318 . . . . 5 (𝑢 = 𝑌 → (¬ 𝑢𝑣 ↔ ¬ 𝑌𝑣))
2726ralbidv 3176 . . . 4 (𝑢 = 𝑌 → (∀𝑣𝑌 ¬ 𝑢𝑣 ↔ ∀𝑣𝑌 ¬ 𝑌𝑣))
2827rspcev 3622 . . 3 (( 𝑌𝑌 ∧ ∀𝑣𝑌 ¬ 𝑌𝑣) → ∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣)
2924, 28mpan2 691 . 2 ( 𝑌𝑌 → ∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣)
3020, 29impbid1 225 1 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣 𝑌𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  wss 3963  wpss 3964   cuni 4912   Or wor 5596   [] crpss 7741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-so 5598  df-xp 5695  df-rel 5696  df-rpss 7742
This theorem is referenced by:  fin2i2  10356  isfin2-2  10357  fin12  10451
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