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Theorem sorpssuni 7441
 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 7438 . . . . . . . . . 10 (( [] Or 𝑌 ∧ (𝑢𝑌𝑣𝑌)) → (𝑢𝑣𝑣𝑢))
21anassrs 471 . . . . . . . . 9 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (𝑢𝑣𝑣𝑢))
3 sspss 4027 . . . . . . . . . . 11 (𝑢𝑣 ↔ (𝑢𝑣𝑢 = 𝑣))
4 orel1 886 . . . . . . . . . . . 12 𝑢𝑣 → ((𝑢𝑣𝑢 = 𝑣) → 𝑢 = 𝑣))
5 eqimss2 3972 . . . . . . . . . . . 12 (𝑢 = 𝑣𝑣𝑢)
64, 5syl6com 37 . . . . . . . . . . 11 ((𝑢𝑣𝑢 = 𝑣) → (¬ 𝑢𝑣𝑣𝑢))
73, 6sylbi 220 . . . . . . . . . 10 (𝑢𝑣 → (¬ 𝑢𝑣𝑣𝑢))
8 ax-1 6 . . . . . . . . . 10 (𝑣𝑢 → (¬ 𝑢𝑣𝑣𝑢))
97, 8jaoi 854 . . . . . . . . 9 ((𝑢𝑣𝑣𝑢) → (¬ 𝑢𝑣𝑣𝑢))
102, 9syl 17 . . . . . . . 8 ((( [] Or 𝑌𝑢𝑌) ∧ 𝑣𝑌) → (¬ 𝑢𝑣𝑣𝑢))
1110ralimdva 3144 . . . . . . 7 (( [] Or 𝑌𝑢𝑌) → (∀𝑣𝑌 ¬ 𝑢𝑣 → ∀𝑣𝑌 𝑣𝑢))
12113impia 1114 . . . . . 6 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → ∀𝑣𝑌 𝑣𝑢)
13 unissb 4833 . . . . . 6 ( 𝑌𝑢 ↔ ∀𝑣𝑌 𝑣𝑢)
1412, 13sylibr 237 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑌𝑢)
15 elssuni 4831 . . . . . 6 (𝑢𝑌𝑢 𝑌)
16153ad2ant2 1131 . . . . 5 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑢 𝑌)
1714, 16eqssd 3932 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑌 = 𝑢)
18 simp2 1134 . . . 4 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑢𝑌)
1917, 18eqeltrd 2890 . . 3 (( [] Or 𝑌𝑢𝑌 ∧ ∀𝑣𝑌 ¬ 𝑢𝑣) → 𝑌𝑌)
2019rexlimdv3a 3245 . 2 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣 𝑌𝑌))
21 elssuni 4831 . . . . 5 (𝑣𝑌𝑣 𝑌)
22 ssnpss 4031 . . . . 5 (𝑣 𝑌 → ¬ 𝑌𝑣)
2321, 22syl 17 . . . 4 (𝑣𝑌 → ¬ 𝑌𝑣)
2423rgen 3116 . . 3 𝑣𝑌 ¬ 𝑌𝑣
25 psseq1 4015 . . . . . 6 (𝑢 = 𝑌 → (𝑢𝑣 𝑌𝑣))
2625notbid 321 . . . . 5 (𝑢 = 𝑌 → (¬ 𝑢𝑣 ↔ ¬ 𝑌𝑣))
2726ralbidv 3162 . . . 4 (𝑢 = 𝑌 → (∀𝑣𝑌 ¬ 𝑢𝑣 ↔ ∀𝑣𝑌 ¬ 𝑌𝑣))
2827rspcev 3571 . . 3 (( 𝑌𝑌 ∧ ∀𝑣𝑌 ¬ 𝑌𝑣) → ∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣)
2924, 28mpan2 690 . 2 ( 𝑌𝑌 → ∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣)
3020, 29impbid1 228 1 ( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣 𝑌𝑌))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881   ⊊ wpss 3882  ∪ cuni 4801   Or wor 5438   [⊊] crpss 7431 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-so 5440  df-xp 5526  df-rel 5527  df-rpss 7432 This theorem is referenced by:  fin2i2  9732  isfin2-2  9733  fin12  9827
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