Theorem suplub2 9490

Description: Bidirectional form of suplub 9489. (Contributed by Mario Carneiro, 6-Sep-2014.)

Hypotheses

Ref	Expression
supmo.1	⊢ (𝜑 → 𝑅 Or 𝐴)
supcl.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
suplub2.3	⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Assertion

Ref	Expression
suplub2	⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))

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

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)   𝐶(𝑥, 𝑦)

Proof of Theorem suplub2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		supmo.1	. . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴)
2		supcl.2	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
3	1, 2	suplub 9489	. . 3 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
4	3	expdimp 458	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
5		breq2 5112	. . . 4 ⊢ (𝑧 = 𝑤 → (𝐶𝑅𝑧 ↔ 𝐶𝑅𝑤))
6	5	cbvrexvw 3245	. . 3 ⊢ (∃𝑧 ∈ 𝐵 𝐶𝑅𝑧 ↔ ∃𝑤 ∈ 𝐵 𝐶𝑅𝑤)
7		breq2 5112	. . . . . . 7 ⊢ (sup(𝐵, 𝐴, 𝑅) = 𝑤 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅𝑤))
8	7	biimprd 251	. . . . . 6 ⊢ (sup(𝐵, 𝐴, 𝑅) = 𝑤 → (𝐶𝑅𝑤 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
9	8	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (sup(𝐵, 𝐴, 𝑅) = 𝑤 → (𝐶𝑅𝑤 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅))))
10	1	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → 𝑅 Or 𝐴)
11		simplr 781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → 𝐶 ∈ 𝐴)
12		suplub2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
13	12	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
14	13	sselda 3913	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐴)
15	1, 2	supcl 9487	. . . . . . . 8 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
16	15	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
17		sotr 5591	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝐶 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴 ∧ sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)) → ((𝐶𝑅𝑤 ∧ 𝑤𝑅sup(𝐵, 𝐴, 𝑅)) → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
18	10, 11, 14, 16, 17	syl13anc 1400	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ((𝐶𝑅𝑤 ∧ 𝑤𝑅sup(𝐵, 𝐴, 𝑅)) → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
19	18	expcomd 422	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → (𝐶𝑅𝑤 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅))))
20	1, 2	supub 9488	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
21	20	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝑤 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
22	21	imp 412	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤)
23		sotric 5596	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ (sup(𝐵, 𝐴, 𝑅) = 𝑤 ∨ 𝑤𝑅sup(𝐵, 𝐴, 𝑅))))
24	10, 16, 14, 23	syl12anc 850	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ (sup(𝐵, 𝐴, 𝑅) = 𝑤 ∨ 𝑤𝑅sup(𝐵, 𝐴, 𝑅))))
25	24	con2bid 357	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ((sup(𝐵, 𝐴, 𝑅) = 𝑤 ∨ 𝑤𝑅sup(𝐵, 𝐴, 𝑅)) ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
26	22, 25	mpbird 260	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (sup(𝐵, 𝐴, 𝑅) = 𝑤 ∨ 𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
27	9, 19, 26	mpjaod 874	. . . 4 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (𝐶𝑅𝑤 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
28	27	rexlimdva 3167	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∃𝑤 ∈ 𝐵 𝐶𝑅𝑤 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
29	6, 28	biimtrid 245	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∃𝑧 ∈ 𝐵 𝐶𝑅𝑧 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
30	4, 29	impbid 215	1 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   class class class wbr 5108   Or wor 5565  supcsup 9469

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-po 5566  df-so 5567  df-iota 6520  df-riota 7408  df-sup 9471

This theorem is used by:  infglbb  9521  suprlub  12321  supxrlub  13495