Theorem supsnti 7106

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 3661	. . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ {𝐵})
4	2, 3	syl 14	. 2 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
5		eqid 2204	. . . . . 6 ⊢ 𝐵 = 𝐵
6	1	ralrimivva 2587	. . . . . . 7 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
7		eqeq1 2211	. . . . . . . . . 10 ⊢ (𝑢 = 𝐵 → (𝑢 = 𝑣 ↔ 𝐵 = 𝑣))
8		breq1 4046	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐵 → (𝑢𝑅𝑣 ↔ 𝐵𝑅𝑣))
9	8	notbid 668	. . . . . . . . . . 11 ⊢ (𝑢 = 𝐵 → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝐵𝑅𝑣))
10		breq2 4047	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐵 → (𝑣𝑅𝑢 ↔ 𝑣𝑅𝐵))
11	10	notbid 668	. . . . . . . . . . 11 ⊢ (𝑢 = 𝐵 → (¬ 𝑣𝑅𝑢 ↔ ¬ 𝑣𝑅𝐵))
12	9, 11	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑢 = 𝐵 → ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵)))
13	7, 12	bibi12d 235	. . . . . . . . 9 ⊢ (𝑢 = 𝐵 → ((𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ (𝐵 = 𝑣 ↔ (¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵))))
14		eqeq2 2214	. . . . . . . . . 10 ⊢ (𝑣 = 𝐵 → (𝐵 = 𝑣 ↔ 𝐵 = 𝐵))
15		breq2 4047	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐵 → (𝐵𝑅𝑣 ↔ 𝐵𝑅𝐵))
16	15	notbid 668	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → (¬ 𝐵𝑅𝑣 ↔ ¬ 𝐵𝑅𝐵))
17		breq1 4046	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐵 → (𝑣𝑅𝐵 ↔ 𝐵𝑅𝐵))
18	17	notbid 668	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → (¬ 𝑣𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
19	16, 18	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑣 = 𝐵 → ((¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵) ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵)))
20	14, 19	bibi12d 235	. . . . . . . . 9 ⊢ (𝑣 = 𝐵 → ((𝐵 = 𝑣 ↔ (¬ 𝐵𝑅𝑣 ∧ ¬ 𝑣𝑅𝐵)) ↔ (𝐵 = 𝐵 ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵))))
21	13, 20	rspc2v 2889	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) → (𝐵 = 𝐵 ↔ (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵))))
22	2, 2, 21	syl2anc 411	. . . . . . 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 148	. . . . 5 ⊢ (𝜑 → (¬ 𝐵𝑅𝐵 ∧ ¬ 𝐵𝑅𝐵))
25	24	simpld 112	. . . 4 ⊢ (𝜑 → ¬ 𝐵𝑅𝐵)
26	25	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐵}) → ¬ 𝐵𝑅𝐵)
27		elsni 3650	. . . . . 6 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
28	27	breq2d 4055	. . . . 5 ⊢ (𝑥 ∈ {𝐵} → (𝐵𝑅𝑥 ↔ 𝐵𝑅𝐵))
29	28	notbid 668	. . . 4 ⊢ (𝑥 ∈ {𝐵} → (¬ 𝐵𝑅𝑥 ↔ ¬ 𝐵𝑅𝐵))
30	29	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐵}) → (¬ 𝐵𝑅𝑥 ↔ ¬ 𝐵𝑅𝐵))
31	26, 30	mpbird 167	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐵}) → ¬ 𝐵𝑅𝑥)
32	1, 2, 4, 31	supmaxti 7105	1 ⊢ (𝜑 → sup({𝐵}, 𝐴, 𝑅) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1372   ∈ wcel 2175  ∀wral 2483  {csn 3632   class class class wbr 4043  supcsup 7083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-iota 5231  df-riota 5898  df-sup 7085

This theorem is referenced by:  infsnti  7131