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Theorem sorpssint 7675
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 4929 . . . . . 6 (𝑢𝑌 𝑌𝑢)
213ad2ant2 1135 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑌𝑢)
3 sorpssi 7671 . . . . . . . . . 10 (( [] Or 𝑌 ∧ (𝑢𝑌𝑣𝑌)) → (𝑢𝑣𝑣𝑢))
43anassrs 469 . . . . . . . . 9 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (𝑢𝑣𝑣𝑢))
5 sspss 4064 . . . . . . . . . . 11 (𝑣𝑢 ↔ (𝑣𝑢𝑣 = 𝑢))
6 orel1 888 . . . . . . . . . . . 12 𝑣𝑢 → ((𝑣𝑢𝑣 = 𝑢) → 𝑣 = 𝑢))
7 eqimss2 4006 . . . . . . . . . . . 12 (𝑣 = 𝑢𝑢𝑣)
86, 7syl6com 37 . . . . . . . . . . 11 ((𝑣𝑢𝑣 = 𝑢) → (¬ 𝑣𝑢𝑢𝑣))
95, 8sylbi 216 . . . . . . . . . 10 (𝑣𝑢 → (¬ 𝑣𝑢𝑢𝑣))
109jao1i 857 . . . . . . . . 9 ((𝑢𝑣𝑣𝑢) → (¬ 𝑣𝑢𝑢𝑣))
114, 10syl 17 . . . . . . . 8 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (¬ 𝑣𝑢𝑢𝑣))
1211ralimdva 3165 . . . . . . 7 (( [] Or 𝑌𝑢𝑌) → (∀𝑣𝑌 ¬ 𝑣𝑢 → ∀𝑣𝑌 𝑢𝑣))
13123impia 1118 . . . . . 6 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → ∀𝑣𝑌 𝑢𝑣)
14 ssint 4930 . . . . . 6 (𝑢 𝑌 ↔ ∀𝑣𝑌 𝑢𝑣)
1513, 14sylibr 233 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑢 𝑌)
162, 15eqssd 3966 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑌 = 𝑢)
17 simp2 1138 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑢𝑌)
1816, 17eqeltrd 2838 . . 3 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑌𝑌)
1918rexlimdv3a 3157 . 2 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢 𝑌𝑌))
20 intss1 4929 . . . . 5 (𝑣𝑌 𝑌𝑣)
21 ssnpss 4068 . . . . 5 ( 𝑌𝑣 → ¬ 𝑣 𝑌)
2220, 21syl 17 . . . 4 (𝑣𝑌 → ¬ 𝑣 𝑌)
2322rgen 3067 . . 3 𝑣𝑌 ¬ 𝑣 𝑌
24 psseq2 4053 . . . . . 6 (𝑢 = 𝑌 → (𝑣𝑢𝑣 𝑌))
2524notbid 318 . . . . 5 (𝑢 = 𝑌 → (¬ 𝑣𝑢 ↔ ¬ 𝑣 𝑌))
2625ralbidv 3175 . . . 4 (𝑢 = 𝑌 → (∀𝑣𝑌 ¬ 𝑣𝑢 ↔ ∀𝑣𝑌 ¬ 𝑣 𝑌))
2726rspcev 3584 . . 3 (( 𝑌𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣 𝑌) → ∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢)
2823, 27mpan2 690 . 2 ( 𝑌𝑌 → ∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢)
2919, 28impbid1 224 1 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢 𝑌𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wral 3065  wrex 3074  wss 3915  wpss 3916   cint 4912   Or wor 5549   [] crpss 7664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-int 4913  df-br 5111  df-opab 5173  df-so 5551  df-xp 5644  df-rel 5645  df-rpss 7665
This theorem is referenced by:  fin2i2  10261  isfin2-2  10262
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