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Theorem sorpssuni 7718
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 7715 . . . . . . . . . 10 (( [] Or 𝑌 ∧ (𝑢𝑌𝑣𝑌)) → (𝑢𝑣𝑣𝑢))
21anassrs 468 . . . . . . . . 9 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (𝑢𝑣𝑣𝑢))
3 sspss 4098 . . . . . . . . . . 11 (𝑢𝑣 ↔ (𝑢𝑣𝑢 = 𝑣))
4 orel1 887 . . . . . . . . . . . 12 𝑢𝑣 → ((𝑢𝑣𝑢 = 𝑣) → 𝑢 = 𝑣))
5 eqimss2 4040 . . . . . . . . . . . 12 (𝑢 = 𝑣𝑣𝑢)
64, 5syl6com 37 . . . . . . . . . . 11 ((𝑢𝑣𝑢 = 𝑣) → (¬ 𝑢𝑣𝑣𝑢))
73, 6sylbi 216 . . . . . . . . . 10 (𝑢𝑣 → (¬ 𝑢𝑣𝑣𝑢))
8 ax-1 6 . . . . . . . . . 10 (𝑣𝑢 → (¬ 𝑢𝑣𝑣𝑢))
97, 8jaoi 855 . . . . . . . . 9 ((𝑢𝑣𝑣𝑢) → (¬ 𝑢𝑣𝑣𝑢))
102, 9syl 17 . . . . . . . 8 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (¬ 𝑢𝑣𝑣𝑢))
1110ralimdva 3167 . . . . . . 7 (( [] Or 𝑌𝑢𝑌) → (∀𝑣𝑌 ¬ 𝑢𝑣 → ∀𝑣𝑌 𝑣𝑢))
12113impia 1117 . . . . . 6 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → ∀𝑣𝑌 𝑣𝑢)
13 unissb 4942 . . . . . 6 ( 𝑌𝑢 ↔ ∀𝑣𝑌 𝑣𝑢)
1412, 13sylibr 233 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑌𝑢)
15 elssuni 4940 . . . . . 6 (𝑢𝑌𝑢 𝑌)
16153ad2ant2 1134 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑢 𝑌)
1714, 16eqssd 3998 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑌 = 𝑢)
18 simp2 1137 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑢𝑌)
1917, 18eqeltrd 2833 . . 3 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑌𝑌)
2019rexlimdv3a 3159 . 2 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣 𝑌𝑌))
21 elssuni 4940 . . . . 5 (𝑣𝑌𝑣 𝑌)
22 ssnpss 4102 . . . . 5 (𝑣 𝑌 → ¬ 𝑌𝑣)
2321, 22syl 17 . . . 4 (𝑣𝑌 → ¬ 𝑌𝑣)
2423rgen 3063 . . 3 𝑣𝑌 ¬ 𝑌𝑣
25 psseq1 4086 . . . . . 6 (𝑢 = 𝑌 → (𝑢𝑣 𝑌𝑣))
2625notbid 317 . . . . 5 (𝑢 = 𝑌 → (¬ 𝑢𝑣 ↔ ¬ 𝑌𝑣))
2726ralbidv 3177 . . . 4 (𝑢 = 𝑌 → (∀𝑣𝑌 ¬ 𝑢𝑣 ↔ ∀𝑣𝑌 ¬ 𝑌𝑣))
2827rspcev 3612 . . 3 (( 𝑌𝑌 ∧ ∀𝑣𝑌 ¬ 𝑌𝑣) → ∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣)
2924, 28mpan2 689 . 2 ( 𝑌𝑌 → ∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣)
3020, 29impbid1 224 1 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣 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   cuni 4907   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-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-uni 4908  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  fin12  10404
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