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Theorem sorpssint 7719
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 4966 . . . . . 6 (𝑢𝑌 𝑌𝑢)
213ad2ant2 1134 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑌𝑢)
3 sorpssi 7715 . . . . . . . . . 10 (( [] Or 𝑌 ∧ (𝑢𝑌𝑣𝑌)) → (𝑢𝑣𝑣𝑢))
43anassrs 468 . . . . . . . . 9 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (𝑢𝑣𝑣𝑢))
5 sspss 4098 . . . . . . . . . . 11 (𝑣𝑢 ↔ (𝑣𝑢𝑣 = 𝑢))
6 orel1 887 . . . . . . . . . . . 12 𝑣𝑢 → ((𝑣𝑢𝑣 = 𝑢) → 𝑣 = 𝑢))
7 eqimss2 4040 . . . . . . . . . . . 12 (𝑣 = 𝑢𝑢𝑣)
86, 7syl6com 37 . . . . . . . . . . 11 ((𝑣𝑢𝑣 = 𝑢) → (¬ 𝑣𝑢𝑢𝑣))
95, 8sylbi 216 . . . . . . . . . 10 (𝑣𝑢 → (¬ 𝑣𝑢𝑢𝑣))
109jao1i 856 . . . . . . . . 9 ((𝑢𝑣𝑣𝑢) → (¬ 𝑣𝑢𝑢𝑣))
114, 10syl 17 . . . . . . . 8 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (¬ 𝑣𝑢𝑢𝑣))
1211ralimdva 3167 . . . . . . 7 (( [] Or 𝑌𝑢𝑌) → (∀𝑣𝑌 ¬ 𝑣𝑢 → ∀𝑣𝑌 𝑢𝑣))
13123impia 1117 . . . . . 6 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → ∀𝑣𝑌 𝑢𝑣)
14 ssint 4967 . . . . . 6 (𝑢 𝑌 ↔ ∀𝑣𝑌 𝑢𝑣)
1513, 14sylibr 233 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑢 𝑌)
162, 15eqssd 3998 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑌 = 𝑢)
17 simp2 1137 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑢𝑌)
1816, 17eqeltrd 2833 . . 3 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣𝑢) → 𝑌𝑌)
1918rexlimdv3a 3159 . 2 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢 𝑌𝑌))
20 intss1 4966 . . . . 5 (𝑣𝑌 𝑌𝑣)
21 ssnpss 4102 . . . . 5 ( 𝑌𝑣 → ¬ 𝑣 𝑌)
2220, 21syl 17 . . . 4 (𝑣𝑌 → ¬ 𝑣 𝑌)
2322rgen 3063 . . 3 𝑣𝑌 ¬ 𝑣 𝑌
24 psseq2 4087 . . . . . 6 (𝑢 = 𝑌 → (𝑣𝑢𝑣 𝑌))
2524notbid 317 . . . . 5 (𝑢 = 𝑌 → (¬ 𝑣𝑢 ↔ ¬ 𝑣 𝑌))
2625ralbidv 3177 . . . 4 (𝑢 = 𝑌 → (∀𝑣𝑌 ¬ 𝑣𝑢 ↔ ∀𝑣𝑌 ¬ 𝑣 𝑌))
2726rspcev 3612 . . 3 (( 𝑌𝑌 ∧ ∀𝑣𝑌 ¬ 𝑣 𝑌) → ∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢)
2823, 27mpan2 689 . 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 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wral 3061  wrex 3070  wss 3947  wpss 3948   cint 4949   Or wor 5586   [] crpss 7708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-int 4950  df-br 5148  df-opab 5210  df-so 5588  df-xp 5681  df-rel 5682  df-rpss 7709
This theorem is referenced by:  fin2i2  10309  isfin2-2  10310
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