Theorem lub0N 39623

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 2735	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2735	. . 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 4330	. . . 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 39618	. . 3 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾))
11		ral0 4428	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧
12	11	a1bi 362	. . . . . 6 ⊢ (𝑥(le‘𝐾)𝑧 ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))
13	12	ralbii 3081	. . . . 5 ⊢ (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))
14		ral0 4428	. . . . . 6 ⊢ ∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥
15	14	biantrur 530	. . . . 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 480	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → 0 ∈ (Base‘𝐾))
18		breq2 5078	. . . . . . . 8 ⊢ (𝑧 = 0 → (𝑥(le‘𝐾)𝑧 ↔ 𝑥(le‘𝐾) 0 ))
19	18	rspcv 3558	. . . . . . 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 39621	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥(le‘𝐾) 0 ↔ 𝑥 = 0 ))
22	20, 21	sylibd 239	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 → 𝑥 = 0 ))
23	1, 2, 9	op0le 39620	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑧 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑧)
24	23	adantlr 716	. . . . . . . . 9 ⊢ (((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑧)
25	24	ex 412	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → 0 (le‘𝐾)𝑧))
26		breq1 5077	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥(le‘𝐾)𝑧 ↔ 0 (le‘𝐾)𝑧))
27	26	biimprcd 250	. . . . . . . 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 3133	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 0 → ∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧))
31	22, 30	impbid 212	. . . 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 7342	. 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 2770	1 ⊢ (𝐾 ∈ OP → ( 1 ‘∅) = 0 )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3049   ⊆ wss 3885  ∅c0 4263   class class class wbr 5074  ‘cfv 6487  ℩crio 7312  Basecbs 17168  lecple 17216  lubclub 18264  0.cp0 18376  OPcops 39606

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7313  df-ov 7359  df-proset 18249  df-poset 18268  df-lub 18299  df-glb 18300  df-p0 18378  df-oposet 39610

This theorem is referenced by: (None)