Theorem supelti 6415

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 3059	. . . . . 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 61	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
6	1, 5	supclti 6411	. . . 4 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
7		elisset 2613	. . . 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 2083	. . . 4 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, 𝑅) = 𝑥)
10	9	exbii 1536	. . 3 ⊢ (∃𝑥 𝑥 = sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑥sup(𝐵, 𝐴, 𝑅) = 𝑥)
11	8, 10	sylib 120	. 2 ⊢ (𝜑 → ∃𝑥sup(𝐵, 𝐴, 𝑅) = 𝑥)
12		simpr 108	. . 3 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → sup(𝐵, 𝐴, 𝑅) = 𝑥)
13	1, 5	supval2ti 6408	. . . . . . . 8 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
14	13	eqeq1d 2089	. . . . . . 7 ⊢ (𝜑 → (sup(𝐵, 𝐴, 𝑅) = 𝑥 ↔ (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) = 𝑥))
15	14	biimpa 290	. . . . . 6 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) = 𝑥)
16	1, 5	supeuti 6407	. . . . . . . 8 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
17		riota1 5506	. . . . . . . 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 270	. . . . . 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 165	. . . . 5 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → (𝑥 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
21	20	simpld 110	. . . 4 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → 𝑥 ∈ 𝐴)
22	2, 3, 16	jca32 303	. . . . 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 112	. . . . 5 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
24		reupick 3248	. . . . 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 559	. . . 4 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ 𝐴))
26	21, 25	mpbird 165	. . 3 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → 𝑥 ∈ 𝐶)
27	12, 26	eqeltrd 2155	. 2 ⊢ ((𝜑 ∧ sup(𝐵, 𝐴, 𝑅) = 𝑥) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐶)
28	11, 27	exlimddv 1819	1 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∀wral 2348  ∃wrex 2349  ∃!wreu 2350   ⊆ wss 2973   class class class wbr 3785  ℩crio 5487  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-in 2979  df-ss 2986  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:  zsupcl  10343