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Theorem sorpssint 7448
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 4882 . . . . . 6 (𝑢𝑌 𝑌𝑢)
213ad2ant2 1126 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑌𝑢)
3 sorpssi 7444 . . . . . . . . . 10 (( [] Or 𝑌 ∧ (𝑢𝑌𝑣𝑌)) → (𝑢𝑣𝑣𝑢))
43anassrs 468 . . . . . . . . 9 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (𝑢𝑣𝑣𝑢))
5 sspss 4073 . . . . . . . . . . 11 (𝑣𝑢 ↔ (𝑣𝑢𝑣 = 𝑢))
6 orel1 882 . . . . . . . . . . . 12 𝑣𝑢 → ((𝑣𝑢𝑣 = 𝑢) → 𝑣 = 𝑢))
7 eqimss2 4021 . . . . . . . . . . . 12 (𝑣 = 𝑢𝑢𝑣)
86, 7syl6com 37 . . . . . . . . . . 11 ((𝑣𝑢𝑣 = 𝑢) → (¬ 𝑣𝑢𝑢𝑣))
95, 8sylbi 218 . . . . . . . . . 10 (𝑣𝑢 → (¬ 𝑣𝑢𝑢𝑣))
109jao1i 852 . . . . . . . . 9 ((𝑢𝑣𝑣𝑢) → (¬ 𝑣𝑢𝑢𝑣))
114, 10syl 17 . . . . . . . 8 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (¬ 𝑣𝑢𝑢𝑣))
1211ralimdva 3174 . . . . . . 7 (( [] Or 𝑌𝑢𝑌) → (∀𝑣𝑌 ¬ 𝑣𝑢 → ∀𝑣𝑌 𝑢𝑣))
13123impia 1109 . . . . . 6 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → ∀𝑣𝑌 𝑢𝑣)
14 ssint 4883 . . . . . 6 (𝑢 𝑌 ↔ ∀𝑣𝑌 𝑢𝑣)
1513, 14sylibr 235 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑢 𝑌)
162, 15eqssd 3981 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑌 = 𝑢)
17 simp2 1129 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑢𝑌)
1816, 17eqeltrd 2910 . . 3 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑌𝑌)
1918rexlimdv3a 3283 . 2 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢 𝑌𝑌))
20 intss1 4882 . . . . 5 (𝑣𝑌 𝑌𝑣)
21 ssnpss 4077 . . . . 5 ( 𝑌𝑣 → ¬ 𝑣 𝑌)
2220, 21syl 17 . . . 4 (𝑣𝑌 → ¬ 𝑣 𝑌)
2322rgen 3145 . . 3 𝑣𝑌 ¬ 𝑣 𝑌
24 psseq2 4062 . . . . . 6 (𝑢 = 𝑌 → (𝑣𝑢𝑣 𝑌))
2524notbid 319 . . . . 5 (𝑢 = 𝑌 → (¬ 𝑣𝑢 ↔ ¬ 𝑣 𝑌))
2625ralbidv 3194 . . . 4 (𝑢 = 𝑌 → (∀𝑣𝑌 ¬ 𝑣𝑢 ↔ ∀𝑣𝑌 ¬ 𝑣 𝑌))
2726rspcev 3620 . . 3 (( 𝑌𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣 𝑌) → ∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢)
2823, 27mpan2 687 . 2 ( 𝑌𝑌 → ∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢)
2919, 28impbid1 226 1 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢 𝑌𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  wss 3933  wpss 3934   cint 4867   Or wor 5466   [] crpss 7437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-int 4868  df-br 5058  df-opab 5120  df-so 5468  df-xp 5554  df-rel 5555  df-rpss 7438
This theorem is referenced by:  fin2i2  9728  isfin2-2  9729
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