Theorem suplub2 9502

Description: Bidirectional form of suplub 9501. (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 9501	. . 3 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
4	3	expdimp 452	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
5		breq2 5146	. . . 4 ⊢ (𝑧 = 𝑤 → (𝐶𝑅𝑧 ↔ 𝐶𝑅𝑤))
6	5	cbvrexvw 3237	. . 3 ⊢ (∃𝑧 ∈ 𝐵 𝐶𝑅𝑧 ↔ ∃𝑤 ∈ 𝐵 𝐶𝑅𝑤)
7		breq2 5146	. . . . . . 7 ⊢ (sup(𝐵, 𝐴, 𝑅) = 𝑤 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅𝑤))
8	7	biimprd 248	. . . . . 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 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → 𝑅 Or 𝐴)
11		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → 𝐶 ∈ 𝐴)
12		suplub2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
14	13	sselda 3982	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐴)
15	1, 2	supcl 9499	. . . . . . . 8 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
16	15	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
17		sotr 5616	. . . . . . 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 1373	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ((𝐶𝑅𝑤 ∧ 𝑤𝑅sup(𝐵, 𝐴, 𝑅)) → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
19	18	expcomd 416	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → (𝐶𝑅𝑤 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅))))
20	1, 2	supub 9500	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
21	20	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝑤 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
22	21	imp 406	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤)
23		sotric 5621	. . . . . . . 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 836	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ (sup(𝐵, 𝐴, 𝑅) = 𝑤 ∨ 𝑤𝑅sup(𝐵, 𝐴, 𝑅))))
25	24	con2bid 354	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ((sup(𝐵, 𝐴, 𝑅) = 𝑤 ∨ 𝑤𝑅sup(𝐵, 𝐴, 𝑅)) ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
26	22, 25	mpbird 257	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (sup(𝐵, 𝐴, 𝑅) = 𝑤 ∨ 𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
27	9, 19, 26	mpjaod 860	. . . 4 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (𝐶𝑅𝑤 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
28	27	rexlimdva 3154	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∃𝑤 ∈ 𝐵 𝐶𝑅𝑤 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
29	6, 28	biimtrid 242	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∃𝑧 ∈ 𝐵 𝐶𝑅𝑧 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
30	4, 29	impbid 212	1 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107  ∀wral 3060  ∃wrex 3069   ⊆ wss 3950   class class class wbr 5142   Or wor 5590  supcsup 9481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-po 5591  df-so 5592  df-iota 6513  df-riota 7389  df-sup 9483

This theorem is referenced by:  infglbb  9532  suprlub  12233  supxrlub  13368