Theorem suplub2 9220

Description: Bidirectional form of suplub 9219. (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 9219	. . 3 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
4	3	expdimp 453	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
5		breq2 5078	. . . 4 ⊢ (𝑧 = 𝑤 → (𝐶𝑅𝑧 ↔ 𝐶𝑅𝑤))
6	5	cbvrexvw 3384	. . 3 ⊢ (∃𝑧 ∈ 𝐵 𝐶𝑅𝑧 ↔ ∃𝑤 ∈ 𝐵 𝐶𝑅𝑤)
7		breq2 5078	. . . . . . 7 ⊢ (sup(𝐵, 𝐴, 𝑅) = 𝑤 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅𝑤))
8	7	biimprd 247	. . . . . 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 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → 𝑅 Or 𝐴)
11		simplr 766	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → 𝐶 ∈ 𝐴)
12		suplub2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
13	12	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
14	13	sselda 3921	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐴)
15	1, 2	supcl 9217	. . . . . . . 8 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
16	15	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
17		sotr 5527	. . . . . . 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 1371	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ((𝐶𝑅𝑤 ∧ 𝑤𝑅sup(𝐵, 𝐴, 𝑅)) → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
19	18	expcomd 417	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → (𝐶𝑅𝑤 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅))))
20	1, 2	supub 9218	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
21	20	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝑤 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
22	21	imp 407	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤)
23		sotric 5531	. . . . . . . 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 834	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ (sup(𝐵, 𝐴, 𝑅) = 𝑤 ∨ 𝑤𝑅sup(𝐵, 𝐴, 𝑅))))
25	24	con2bid 355	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ((sup(𝐵, 𝐴, 𝑅) = 𝑤 ∨ 𝑤𝑅sup(𝐵, 𝐴, 𝑅)) ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
26	22, 25	mpbird 256	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (sup(𝐵, 𝐴, 𝑅) = 𝑤 ∨ 𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
27	9, 19, 26	mpjaod 857	. . . 4 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (𝐶𝑅𝑤 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
28	27	rexlimdva 3213	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∃𝑤 ∈ 𝐵 𝐶𝑅𝑤 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
29	6, 28	syl5bi 241	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∃𝑧 ∈ 𝐵 𝐶𝑅𝑧 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
30	4, 29	impbid 211	1 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887   class class class wbr 5074   Or wor 5502  supcsup 9199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-po 5503  df-so 5504  df-iota 6391  df-riota 7232  df-sup 9201

This theorem is referenced by:  infglbb  9250  suprlub  11939  supxrlub  13059