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Theorem sorpssint 6845
 Description: In a chain of sets, a minimal element is the intersection of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
sorpssint ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢 𝑌𝑌))
Distinct variable group:   𝑢,𝑌,𝑣

Proof of Theorem sorpssint
StepHypRef Expression
1 intss1 4427 . . . . . 6 (𝑢𝑌 𝑌𝑢)
213ad2ant2 1076 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑌𝑢)
3 sorpssi 6841 . . . . . . . . . 10 (( [] Or 𝑌 ∧ (𝑢𝑌𝑣𝑌)) → (𝑢𝑣𝑣𝑢))
43anassrs 678 . . . . . . . . 9 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (𝑢𝑣𝑣𝑢))
5 ax-1 6 . . . . . . . . . 10 (𝑢𝑣 → (¬ 𝑣𝑢𝑢𝑣))
6 sspss 3668 . . . . . . . . . . 11 (𝑣𝑢 ↔ (𝑣𝑢𝑣 = 𝑢))
7 orel1 396 . . . . . . . . . . . 12 𝑣𝑢 → ((𝑣𝑢𝑣 = 𝑢) → 𝑣 = 𝑢))
8 eqimss2 3621 . . . . . . . . . . . 12 (𝑣 = 𝑢𝑢𝑣)
97, 8syl6com 36 . . . . . . . . . . 11 ((𝑣𝑢𝑣 = 𝑢) → (¬ 𝑣𝑢𝑢𝑣))
106, 9sylbi 206 . . . . . . . . . 10 (𝑣𝑢 → (¬ 𝑣𝑢𝑢𝑣))
115, 10jaoi 393 . . . . . . . . 9 ((𝑢𝑣𝑣𝑢) → (¬ 𝑣𝑢𝑢𝑣))
124, 11syl 17 . . . . . . . 8 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (¬ 𝑣𝑢𝑢𝑣))
1312ralimdva 2945 . . . . . . 7 (( [] Or 𝑌𝑢𝑌) → (∀𝑣𝑌 ¬ 𝑣𝑢 → ∀𝑣𝑌 𝑢𝑣))
14133impia 1253 . . . . . 6 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → ∀𝑣𝑌 𝑢𝑣)
15 ssint 4428 . . . . . 6 (𝑢 𝑌 ↔ ∀𝑣𝑌 𝑢𝑣)
1614, 15sylibr 223 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑢 𝑌)
172, 16eqssd 3585 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑌 = 𝑢)
18 simp2 1055 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑢𝑌)
1917, 18eqeltrd 2688 . . 3 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑌𝑌)
2019rexlimdv3a 3015 . 2 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢 𝑌𝑌))
21 intss1 4427 . . . . 5 (𝑣𝑌 𝑌𝑣)
22 ssnpss 3672 . . . . 5 ( 𝑌𝑣 → ¬ 𝑣 𝑌)
2321, 22syl 17 . . . 4 (𝑣𝑌 → ¬ 𝑣 𝑌)
2423rgen 2906 . . 3 𝑣𝑌 ¬ 𝑣 𝑌
25 psseq2 3657 . . . . . 6 (𝑢 = 𝑌 → (𝑣𝑢𝑣 𝑌))
2625notbid 307 . . . . 5 (𝑢 = 𝑌 → (¬ 𝑣𝑢 ↔ ¬ 𝑣 𝑌))
2726ralbidv 2969 . . . 4 (𝑢 = 𝑌 → (∀𝑣𝑌 ¬ 𝑣𝑢 ↔ ∀𝑣𝑌 ¬ 𝑣 𝑌))
2827rspcev 3282 . . 3 (( 𝑌𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣 𝑌) → ∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢)
2924, 28mpan2 703 . 2 ( 𝑌𝑌 → ∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢)
3020, 29impbid1 214 1 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢 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  ∩ cint 4410   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-int 4411  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
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