Theorem supelti 6882

Description: Supremum membership in a set. (Contributed by Jim Kingdon, 16-Jan-2022.)

Hypotheses

Ref	Expression
supelti.ti	⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
supelti.ex	⊢ (𝜑 → ∃𝑥 ∈ 𝐶 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
supelti.ss	⊢ (𝜑 → 𝐶 ⊆ 𝐴)

Assertion

Ref	Expression
supelti	⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐶)

Distinct variable groups:   𝑢,𝐴,𝑣,𝑥   𝑦,𝐴,𝑧,𝑥   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶   𝑢,𝑅,𝑣,𝑥   𝑦,𝑅,𝑧   𝜑,𝑢,𝑣,𝑥

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐵(𝑣,𝑢)   𝐶(𝑦,𝑧,𝑣,𝑢)

Proof of Theorem supelti

Step	Hyp	Ref	Expression
1		supelti.ti	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
2		supelti.ss	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
3		supelti.ex	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ 𝐶 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
4		ssrexv 3157	. . . . . 6 ⊢ (𝐶 ⊆ 𝐴 → (∃𝑥 ∈ 𝐶 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
5	2, 3, 4	sylc 62	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
6	1, 5	supclti 6878	. . . 4 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
7		elisset 2695	. . . 4 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐴 → ∃𝑥 𝑥 = sup(𝐵, 𝐴, 𝑅))
8	6, 7	syl 14	. . 3 ⊢ (𝜑 → ∃𝑥 𝑥 = sup(𝐵, 𝐴, 𝑅))
9		eqcom 2139	. . . 4 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, 𝑅) = 𝑥)
10	9	exbii 1584	. . 3 ⊢ (∃𝑥 𝑥 = sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑥sup(𝐵, 𝐴, 𝑅) = 𝑥)
11	8, 10	sylib 121	. 2 ⊢ (𝜑 → ∃𝑥sup(𝐵, 𝐴, 𝑅) = 𝑥)
12		simpr 109	. . 3 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → sup(𝐵, 𝐴, 𝑅) = 𝑥)
13	1, 5	supval2ti 6875	. . . . . . . 8 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
14	13	eqeq1d 2146	. . . . . . 7 ⊢ (𝜑 → (sup(𝐵, 𝐴, 𝑅) = 𝑥 ↔ (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) = 𝑥))
15	14	biimpa 294	. . . . . 6 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) = 𝑥)
16	1, 5	supeuti 6874	. . . . . . . 8 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
17		riota1 5741	. . . . . . . 8 ⊢ (∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → ((𝑥 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ↔ (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) = 𝑥))
18	16, 17	syl 14	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ↔ (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) = 𝑥))
19	18	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → ((𝑥 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ↔ (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) = 𝑥))
20	15, 19	mpbird 166	. . . . 5 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → (𝑥 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
21	20	simpld 111	. . . 4 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → 𝑥 ∈ 𝐴)
22	2, 3, 16	jca32 308	. . . . 5 ⊢ (𝜑 → (𝐶 ⊆ 𝐴 ∧ (∃𝑥 ∈ 𝐶 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ∧ ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))))
23	20	simprd 113	. . . . 5 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
24		reupick 3355	. . . . 5 ⊢ (((𝐶 ⊆ 𝐴 ∧ (∃𝑥 ∈ 𝐶 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ∧ ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))) ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ 𝐴))
25	22, 23, 24	syl2an2r 584	. . . 4 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ 𝐴))
26	21, 25	mpbird 166	. . 3 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → 𝑥 ∈ 𝐶)
27	12, 26	eqeltrd 2214	. 2 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐶)
28	11, 27	exlimddv 1870	1 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2414  ∃wrex 2415  ∃!wreu 2416   ⊆ wss 3066   class class class wbr 3924  ℩crio 5722  supcsup 6862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-riota 5723  df-sup 6864

This theorem is referenced by:  zsupcl  11629