Theorem lub0N 38059

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.

Step	Hyp	Ref	Expression
1		eqid 2733	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2733	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		lub0.u	. . 3 ⊢ 1 = (lub‘𝐾)
4		biid 261	. . 3 ⊢ ((∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
5		id 22	. . 3 ⊢ (𝐾 ∈ OP → 𝐾 ∈ OP)
6		0ss 4397	. . . 4 ⊢ ∅ ⊆ (Base‘𝐾)
7	6	a1i 11	. . 3 ⊢ (𝐾 ∈ OP → ∅ ⊆ (Base‘𝐾))
8	1, 2, 3, 4, 5, 7	lubval 18309	. 2 ⊢ (𝐾 ∈ OP → ( 1 ‘∅) = (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))))
9		lub0.z	. . . 4 ⊢ 0 = (0.‘𝐾)
10	1, 9	op0cl 38054	. . 3 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾))
11		ral0 4513	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧
12	11	a1bi 363	. . . . . 6 ⊢ (𝑥(le‘𝐾)𝑧 ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))
13	12	ralbii 3094	. . . . 5 ⊢ (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))
14		ral0 4513	. . . . . 6 ⊢ ∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥
15	14	biantrur 532	. . . . 5 ⊢ (∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧) ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
16	13, 15	bitri 275	. . . 4 ⊢ (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
17	10	adantr 482	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → 0 ∈ (Base‘𝐾))
18		breq2 5153	. . . . . . . 8 ⊢ (𝑧 = 0 → (𝑥(le‘𝐾)𝑧 ↔ 𝑥(le‘𝐾) 0 ))
19	18	rspcv 3609	. . . . . . 7 ⊢ ( 0 ∈ (Base‘𝐾) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 → 𝑥(le‘𝐾) 0 ))
20	17, 19	syl 17	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 → 𝑥(le‘𝐾) 0 ))
21	1, 2, 9	ople0 38057	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥(le‘𝐾) 0 ↔ 𝑥 = 0 ))
22	20, 21	sylibd 238	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 → 𝑥 = 0 ))
23	1, 2, 9	op0le 38056	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑧 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑧)
24	23	adantlr 714	. . . . . . . . 9 ⊢ (((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑧)
25	24	ex 414	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → 0 (le‘𝐾)𝑧))
26		breq1 5152	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥(le‘𝐾)𝑧 ↔ 0 (le‘𝐾)𝑧))
27	26	biimprcd 249	. . . . . . . 8 ⊢ ( 0 (le‘𝐾)𝑧 → (𝑥 = 0 → 𝑥(le‘𝐾)𝑧))
28	25, 27	syl6 35	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → (𝑥 = 0 → 𝑥(le‘𝐾)𝑧)))
29	28	com23 86	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 0 → (𝑧 ∈ (Base‘𝐾) → 𝑥(le‘𝐾)𝑧)))
30	29	ralrimdv 3153	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 0 → ∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧))
31	22, 30	impbid 211	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 ↔ 𝑥 = 0 ))
32	16, 31	bitr3id 285	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → ((∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)) ↔ 𝑥 = 0 ))
33	10, 32	riota5 7395	. 2 ⊢ (𝐾 ∈ OP → (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))) = 0 )
34	8, 33	eqtrd 2773	1 ⊢ (𝐾 ∈ OP → ( 1 ‘∅) = 0 )

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062   ⊆ wss 3949  ∅c0 4323   class class class wbr 5149  ‘cfv 6544  ℩crio 7364  Basecbs 17144  lecple 17204  lubclub 18262  0.cp0 18376  OPcops 38042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-proset 18248  df-poset 18266  df-lub 18299  df-glb 18300  df-p0 18378  df-oposet 38046

This theorem is referenced by: (None)