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Theorem lub0N 33991
Description: The least upper bound of the empty set is the zero element. (Contributed by NM, 15-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lub0.u 1 = (lub‘𝐾)
lub0.z 0 = (0.‘𝐾)
Assertion
Ref Expression
lub0N (𝐾 ∈ OP → ( 1 ‘∅) = 0 )

Proof of Theorem lub0N
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2621 . . 3 (le‘𝐾) = (le‘𝐾)
3 lub0.u . . 3 1 = (lub‘𝐾)
4 biid 251 . . 3 ((∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))
5 id 22 . . 3 (𝐾 ∈ OP → 𝐾 ∈ OP)
6 0ss 3949 . . . 4 ∅ ⊆ (Base‘𝐾)
76a1i 11 . . 3 (𝐾 ∈ OP → ∅ ⊆ (Base‘𝐾))
81, 2, 3, 4, 5, 7lubval 16916 . 2 (𝐾 ∈ OP → ( 1 ‘∅) = (𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))))
9 lub0.z . . . 4 0 = (0.‘𝐾)
101, 9op0cl 33986 . . 3 (𝐾 ∈ OP → 0 ∈ (Base‘𝐾))
11 ral0 4053 . . . . . . 7 𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧
1211a1bi 352 . . . . . 6 (𝑥(le‘𝐾)𝑧 ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))
1312ralbii 2975 . . . . 5 (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))
14 ral0 4053 . . . . . 6 𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥
1514biantrur 527 . . . . 5 (∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧) ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))
1613, 15bitri 264 . . . 4 (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))
1710adantr 481 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → 0 ∈ (Base‘𝐾))
18 breq2 4622 . . . . . . . 8 (𝑧 = 0 → (𝑥(le‘𝐾)𝑧𝑥(le‘𝐾) 0 ))
1918rspcv 3294 . . . . . . 7 ( 0 ∈ (Base‘𝐾) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧𝑥(le‘𝐾) 0 ))
2017, 19syl 17 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧𝑥(le‘𝐾) 0 ))
211, 2, 9ople0 33989 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥(le‘𝐾) 0𝑥 = 0 ))
2220, 21sylibd 229 . . . . 5 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧𝑥 = 0 ))
231, 2, 9op0le 33988 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑧 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑧)
2423adantlr 750 . . . . . . . . 9 (((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑧)
2524ex 450 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → 0 (le‘𝐾)𝑧))
26 breq1 4621 . . . . . . . . 9 (𝑥 = 0 → (𝑥(le‘𝐾)𝑧0 (le‘𝐾)𝑧))
2726biimprcd 240 . . . . . . . 8 ( 0 (le‘𝐾)𝑧 → (𝑥 = 0𝑥(le‘𝐾)𝑧))
2825, 27syl6 35 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → (𝑥 = 0𝑥(le‘𝐾)𝑧)))
2928com23 86 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 0 → (𝑧 ∈ (Base‘𝐾) → 𝑥(le‘𝐾)𝑧)))
3029ralrimdv 2963 . . . . 5 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 0 → ∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧))
3122, 30impbid 202 . . . 4 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧𝑥 = 0 ))
3216, 31syl5bbr 274 . . 3 ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → ((∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)) ↔ 𝑥 = 0 ))
3310, 32riota5 6597 . 2 (𝐾 ∈ OP → (𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))) = 0 )
348, 33eqtrd 2655 1 (𝐾 ∈ OP → ( 1 ‘∅) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3559  c0 3896   class class class wbr 4618  cfv 5852  crio 6570  Basecbs 15792  lecple 15880  lubclub 16874  0.cp0 16969  OPcops 33974
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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-preset 16860  df-poset 16878  df-lub 16906  df-glb 16907  df-p0 16971  df-oposet 33978
This theorem is referenced by: (None)
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