MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sorpssuni Structured version   Visualization version   GIF version

Theorem sorpssuni 6899
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 6896 . . . . . . . . . 10 (( [] Or 𝑌 ∧ (𝑢𝑌𝑣𝑌)) → (𝑢𝑣𝑣𝑢))
21anassrs 679 . . . . . . . . 9 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (𝑢𝑣𝑣𝑢))
3 sspss 3684 . . . . . . . . . . 11 (𝑢𝑣 ↔ (𝑢𝑣𝑢 = 𝑣))
4 orel1 397 . . . . . . . . . . . 12 𝑢𝑣 → ((𝑢𝑣𝑢 = 𝑣) → 𝑢 = 𝑣))
5 eqimss2 3637 . . . . . . . . . . . 12 (𝑢 = 𝑣𝑣𝑢)
64, 5syl6com 37 . . . . . . . . . . 11 ((𝑢𝑣𝑢 = 𝑣) → (¬ 𝑢𝑣𝑣𝑢))
73, 6sylbi 207 . . . . . . . . . 10 (𝑢𝑣 → (¬ 𝑢𝑣𝑣𝑢))
8 ax-1 6 . . . . . . . . . 10 (𝑣𝑢 → (¬ 𝑢𝑣𝑣𝑢))
97, 8jaoi 394 . . . . . . . . 9 ((𝑢𝑣𝑣𝑢) → (¬ 𝑢𝑣𝑣𝑢))
102, 9syl 17 . . . . . . . 8 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (¬ 𝑢𝑣𝑣𝑢))
1110ralimdva 2956 . . . . . . 7 (( [] Or 𝑌𝑢𝑌) → (∀𝑣𝑌 ¬ 𝑢𝑣 → ∀𝑣𝑌 𝑣𝑢))
12113impia 1258 . . . . . 6 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → ∀𝑣𝑌 𝑣𝑢)
13 unissb 4435 . . . . . 6 ( 𝑌𝑢 ↔ ∀𝑣𝑌 𝑣𝑢)
1412, 13sylibr 224 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑌𝑢)
15 elssuni 4433 . . . . . 6 (𝑢𝑌𝑢 𝑌)
16153ad2ant2 1081 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑢 𝑌)
1714, 16eqssd 3600 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑌 = 𝑢)
18 simp2 1060 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑢𝑌)
1917, 18eqeltrd 2698 . . 3 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑌𝑌)
2019rexlimdv3a 3026 . 2 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣 𝑌𝑌))
21 elssuni 4433 . . . . 5 (𝑣𝑌𝑣 𝑌)
22 ssnpss 3688 . . . . 5 (𝑣 𝑌 → ¬ 𝑌𝑣)
2321, 22syl 17 . . . 4 (𝑣𝑌 → ¬ 𝑌𝑣)
2423rgen 2917 . . 3 𝑣𝑌 ¬ 𝑌𝑣
25 psseq1 3672 . . . . . 6 (𝑢 = 𝑌 → (𝑢𝑣 𝑌𝑣))
2625notbid 308 . . . . 5 (𝑢 = 𝑌 → (¬ 𝑢𝑣 ↔ ¬ 𝑌𝑣))
2726ralbidv 2980 . . . 4 (𝑢 = 𝑌 → (∀𝑣𝑌 ¬ 𝑢𝑣 ↔ ∀𝑣𝑌 ¬ 𝑌𝑣))
2827rspcev 3295 . . 3 (( 𝑌𝑌 ∧ ∀𝑣𝑌 ¬ 𝑌𝑣) → ∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣)
2924, 28mpan2 706 . 2 ( 𝑌𝑌 → ∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣)
3020, 29impbid1 215 1 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣 𝑌𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  wss 3555  wpss 3556   cuni 4402   Or wor 4994   [] crpss 6889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-so 4996  df-xp 5080  df-rel 5081  df-rpss 6890
This theorem is referenced by:  fin2i2  9084  isfin2-2  9085  fin12  9179
  Copyright terms: Public domain W3C validator