Theorem supsnti 7128

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 3667	. . 3 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ {𝐵})
4	2, 3	syl 14	. 2 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
5		eqid 2206	. . . . . 6 ⊢ 𝐵 = 𝐵
6	1	ralrimivva 2589	. . . . . . 7 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
7		eqeq1 2213	. . . . . . . . . 10 ⊢ (𝑢 = 𝐵 → (𝑢 = 𝑣 ↔ 𝐵 = 𝑣))
8		breq1 4057	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐵 → (𝑢𝑅𝑣 ↔ 𝐵𝑅𝑣))
9	8	notbid 669	. . . . . . . . . . 11 ⊢ (𝑢 = 𝐵 → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝐵𝑅𝑣))
10		breq2 4058	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐵 → (𝑣𝑅𝑢 ↔ 𝑣𝑅𝐵))
11	10	notbid 669	. . . . . . . . . . 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 2216	. . . . . . . . . 10 ⊢ (𝑣 = 𝐵 → (𝐵 = 𝑣 ↔ 𝐵 = 𝐵))
15		breq2 4058	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐵 → (𝐵𝑅𝑣 ↔ 𝐵𝑅𝐵))
16	15	notbid 669	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐵 → (¬ 𝐵𝑅𝑣 ↔ ¬ 𝐵𝑅𝐵))
17		breq1 4057	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐵 → (𝑣𝑅𝐵 ↔ 𝐵𝑅𝐵))
18	17	notbid 669	. . . . . . . . . . 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 2894	. . . . . . . 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 3656	. . . . . 6 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
28	27	breq2d 4066	. . . . 5 ⊢ (𝑥 ∈ {𝐵} → (𝐵𝑅𝑥 ↔ 𝐵𝑅𝐵))
29	28	notbid 669	. . . 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 7127	1 ⊢ (𝜑 → sup({𝐵}, 𝐴, 𝑅) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2177  ∀wral 2485  {csn 3638   class class class wbr 4054  supcsup 7105

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-iota 5246  df-riota 5917  df-sup 7107

This theorem is referenced by:  infsnti  7153