Theorem supsnti 6418

Description: The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.)

Hypotheses

Ref	Expression
supsnti.ti	⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
supsnti.b	⊢ (𝜑 → 𝐵 ∈ 𝐴)

Assertion

Ref	Expression
supsnti	⊢ (𝜑 → sup({𝐵}, 𝐴, 𝑅) = 𝐵)

Distinct variable groups:   𝑢,𝐴,𝑣   𝑢,𝐵,𝑣   𝑢,𝑅,𝑣   𝜑,𝑢,𝑣

Proof of Theorem supsnti

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		supsnti.ti	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
2		supsnti.b	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
3		snidg 3423	. . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ {𝐵})
4	2, 3	syl 14	. 2 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
5		eqid 2081	. . . . . 6 ⊢ 𝐵 = 𝐵
6	1	ralrimivva 2443	. . . . . . 7 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
7		eqeq1 2087	. . . . . . . . . 10 ⊢ (𝑢 = 𝐵 → (𝑢 = 𝑣 ↔ 𝐵 = 𝑣))
8		breq1 3788	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐵 → (𝑢𝑅𝑣 ↔ 𝐵𝑅𝑣))
9	8	notbid 624	. . . . . . . . . . 11 ⊢ (𝑢 = 𝐵 → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝐵𝑅𝑣))
10		breq2 3789	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐵 → (𝑣𝑅𝑢 ↔ 𝑣𝑅𝐵))
11	10	notbid 624	. . . . . . . . . . 11 ⊢ (𝑢 = 𝐵 → (¬ 𝑣𝑅𝑢 ↔ ¬ 𝑣𝑅𝐵))
12	9, 11	anbi12d 456	. . . . . . . . . 10 ⊢ (𝑢 = 𝐵 → ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵)))
13	7, 12	bibi12d 233	. . . . . . . . 9 ⊢ (𝑢 = 𝐵 → ((𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ (𝐵 = 𝑣 ↔ (¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵))))
14		eqeq2 2090	. . . . . . . . . 10 ⊢ (𝑣 = 𝐵 → (𝐵 = 𝑣 ↔ 𝐵 = 𝐵))
15		breq2 3789	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐵 → (𝐵𝑅𝑣 ↔ 𝐵𝑅𝐵))
16	15	notbid 624	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → (¬ 𝐵𝑅𝑣 ↔ ¬ 𝐵𝑅𝐵))
17		breq1 3788	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐵 → (𝑣𝑅𝐵 ↔ 𝐵𝑅𝐵))
18	17	notbid 624	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → (¬ 𝑣𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
19	16, 18	anbi12d 456	. . . . . . . . . 10 ⊢ (𝑣 = 𝐵 → ((¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵) ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵)))
20	14, 19	bibi12d 233	. . . . . . . . 9 ⊢ (𝑣 = 𝐵 → ((𝐵 = 𝑣 ↔ (¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵)) ↔ (𝐵 = 𝐵 ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵))))
21	13, 20	rspc2v 2713	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) → (𝐵 = 𝐵 ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵))))
22	2, 2, 21	syl2anc 403	. . . . . . 7 ⊢ (𝜑 → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) → (𝐵 = 𝐵 ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵))))
23	6, 22	mpd 13	. . . . . 6 ⊢ (𝜑 → (𝐵 = 𝐵 ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵)))
24	5, 23	mpbii 146	. . . . 5 ⊢ (𝜑 → (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵))
25	24	simpld 110	. . . 4 ⊢ (𝜑 → ¬ 𝐵𝑅𝐵)
26	25	adantr 270	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐵}) → ¬ 𝐵𝑅𝐵)
27		elsni 3416	. . . . . 6 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
28	27	breq2d 3797	. . . . 5 ⊢ (𝑥 ∈ {𝐵} → (𝐵𝑅𝑥 ↔ 𝐵𝑅𝐵))
29	28	notbid 624	. . . 4 ⊢ (𝑥 ∈ {𝐵} → (¬ 𝐵𝑅𝑥 ↔ ¬ 𝐵𝑅𝐵))
30	29	adantl 271	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐵}) → (¬ 𝐵𝑅𝑥 ↔ ¬ 𝐵𝑅𝐵))
31	26, 30	mpbird 165	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐵}) → ¬ 𝐵𝑅𝑥)
32	1, 2, 4, 31	supmaxti 6417	1 ⊢ (𝜑 → sup({𝐵}, 𝐴, 𝑅) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ∀wral 2348  {csn 3398   class class class wbr 3785  supcsup 6395

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-riota 5488  df-sup 6397

This theorem is referenced by:  infsnti  6443