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Theorem sorpssuni 6844
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 6841 . . . . . . . . . 10 (( [] Or 𝑌 ∧ (𝑢𝑌𝑣𝑌)) → (𝑢𝑣𝑣𝑢))
21anassrs 678 . . . . . . . . 9 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (𝑢𝑣𝑣𝑢))
3 sspss 3668 . . . . . . . . . . 11 (𝑢𝑣 ↔ (𝑢𝑣𝑢 = 𝑣))
4 orel1 396 . . . . . . . . . . . 12 𝑢𝑣 → ((𝑢𝑣𝑢 = 𝑣) → 𝑢 = 𝑣))
5 eqimss2 3621 . . . . . . . . . . . 12 (𝑢 = 𝑣𝑣𝑢)
64, 5syl6com 36 . . . . . . . . . . 11 ((𝑢𝑣𝑢 = 𝑣) → (¬ 𝑢𝑣𝑣𝑢))
73, 6sylbi 206 . . . . . . . . . 10 (𝑢𝑣 → (¬ 𝑢𝑣𝑣𝑢))
8 ax-1 6 . . . . . . . . . 10 (𝑣𝑢 → (¬ 𝑢𝑣𝑣𝑢))
97, 8jaoi 393 . . . . . . . . 9 ((𝑢𝑣𝑣𝑢) → (¬ 𝑢𝑣𝑣𝑢))
102, 9syl 17 . . . . . . . 8 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (¬ 𝑢𝑣𝑣𝑢))
1110ralimdva 2945 . . . . . . 7 (( [] Or 𝑌𝑢𝑌) → (∀𝑣𝑌 ¬ 𝑢𝑣 → ∀𝑣𝑌 𝑣𝑢))
12113impia 1253 . . . . . 6 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → ∀𝑣𝑌 𝑣𝑢)
13 unissb 4405 . . . . . 6 ( 𝑌𝑢 ↔ ∀𝑣𝑌 𝑣𝑢)
1412, 13sylibr 223 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑌𝑢)
15 elssuni 4403 . . . . . 6 (𝑢𝑌𝑢 𝑌)
16153ad2ant2 1076 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑢 𝑌)
1714, 16eqssd 3585 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑌 = 𝑢)
18 simp2 1055 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑢𝑌)
1917, 18eqeltrd 2688 . . 3 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑌𝑌)
2019rexlimdv3a 3015 . 2 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣 𝑌𝑌))
21 elssuni 4403 . . . . 5 (𝑣𝑌𝑣 𝑌)
22 ssnpss 3672 . . . . 5 (𝑣 𝑌 → ¬ 𝑌𝑣)
2321, 22syl 17 . . . 4 (𝑣𝑌 → ¬ 𝑌𝑣)
2423rgen 2906 . . 3 𝑣𝑌 ¬ 𝑌𝑣
25 psseq1 3656 . . . . . 6 (𝑢 = 𝑌 → (𝑢𝑣 𝑌𝑣))
2625notbid 307 . . . . 5 (𝑢 = 𝑌 → (¬ 𝑢𝑣 ↔ ¬ 𝑌𝑣))
2726ralbidv 2969 . . . 4 (𝑢 = 𝑌 → (∀𝑣𝑌 ¬ 𝑢𝑣 ↔ ∀𝑣𝑌 ¬ 𝑌𝑣))
2827rspcev 3282 . . 3 (( 𝑌𝑌 ∧ ∀𝑣𝑌 ¬ 𝑌𝑣) → ∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣)
2924, 28mpan2 703 . 2 ( 𝑌𝑌 → ∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣)
3020, 29impbid1 214 1 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣 𝑌𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897  wss 3540  wpss 3541   cuni 4372   Or wor 4958   [] crpss 6834
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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
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-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-so 4960  df-xp 5044  df-rel 5045  df-rpss 6835
This theorem is referenced by:  fin2i2  9023  isfin2-2  9024  fin12  9118
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