Theorem lub0N 39818

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 2764	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2764	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		lub0.u	. . 3 ⊢ 1 = (lub‘𝐾)
4		biid 263	. . 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 4356	. . . 4 ⊢ ∅ ⊆ (Base‘𝐾)
7	6	a1i 11	. . 3 ⊢ (𝐾 ∈ OP → ∅ ⊆ (Base‘𝐾))
8	1, 2, 3, 4, 5, 7	lubval 18388	. 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 39813	. . 3 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾))
11		ral0 4454	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧
12	11	a1bi 364	. . . . . 6 ⊢ (𝑥(le‘𝐾)𝑧 ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))
13	12	ralbii 3110	. . . . 5 ⊢ (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))
14		ral0 4454	. . . . . 6 ⊢ ∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥
15	14	biantrur 538	. . . . 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 277	. . . 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 484	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → 0 ∈ (Base‘𝐾))
18		breq2 5106	. . . . . . . 8 ⊢ (𝑧 = 0 → (𝑥(le‘𝐾)𝑧 ↔ 𝑥(le‘𝐾) 0 ))
19	18	rspcv 3579	. . . . . . 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 39816	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥(le‘𝐾) 0 ↔ 𝑥 = 0 ))
22	20, 21	sylibd 241	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 → 𝑥 = 0 ))
23	1, 2, 9	op0le 39815	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑧 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑧)
24	23	adantlr 725	. . . . . . . . 9 ⊢ (((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑧)
25	24	ex 416	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → 0 (le‘𝐾)𝑧))
26		breq1 5105	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥(le‘𝐾)𝑧 ↔ 0 (le‘𝐾)𝑧))
27	26	biimprcd 252	. . . . . . . 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 3162	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 0 → ∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧))
31	22, 30	impbid 214	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 ↔ 𝑥 = 0 ))
32	16, 31	bitr3id 287	. . 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 7384	. 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 2799	1 ⊢ (𝐾 ∈ OP → ( 1 ‘∅) = 0 )

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078   ⊆ wss 3906  ∅c0 4287   class class class wbr 5102  ‘cfv 6523  ℩crio 7354  Basecbs 17247  lecple 17295  lubclub 18343  0.cp0 18455  OPcops 39801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-proset 18328  df-poset 18347  df-lub 18378  df-glb 18379  df-p0 18457  df-oposet 39805

This theorem is referenced by: (None)