Theorem suplub2 8367

Description: Bidirectional form of suplub 8366. (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 8366	. . 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 4657	. . . 4 ⊢ (𝑧 = 𝑤 → (𝐶𝑅𝑧 ↔ 𝐶𝑅𝑤))
6	5	cbvrexv 3172	. . 3 ⊢ (∃𝑧 ∈ 𝐵 𝐶𝑅𝑧 ↔ ∃𝑤 ∈ 𝐵 𝐶𝑅𝑤)
7		breq2 4657	. . . . . . 7 ⊢ (sup(𝐵, 𝐴, 𝑅) = 𝑤 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅𝑤))
8	7	biimprd 238	. . . . . 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 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → 𝑅 Or 𝐴)
11		simplr 792	. . . . . . 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 3603	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐴)
15	1, 2	supcl 8364	. . . . . . . 8 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
16	15	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
17		sotr 5057	. . . . . . 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 1328	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ((𝐶𝑅𝑤 ∧ 𝑤𝑅sup(𝐵, 𝐴, 𝑅)) → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
19	18	expcomd 454	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → (𝐶𝑅𝑤 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅))))
20	1, 2	supub 8365	. . . . . . . 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 445	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤)
23		sotric 5061	. . . . . . . 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 1324	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ (sup(𝐵, 𝐴, 𝑅) = 𝑤 ∨ 𝑤𝑅sup(𝐵, 𝐴, 𝑅))))
25	24	con2bid 344	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → ((sup(𝐵, 𝐴, 𝑅) = 𝑤 ∨ 𝑤𝑅sup(𝐵, 𝐴, 𝑅)) ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
26	22, 25	mpbird 247	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (sup(𝐵, 𝐴, 𝑅) = 𝑤 ∨ 𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
27	9, 19, 26	mpjaod 396	. . . 4 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝐴) ∧ 𝑤 ∈ 𝐵) → (𝐶𝑅𝑤 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
28	27	rexlimdva 3031	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∃𝑤 ∈ 𝐵 𝐶𝑅𝑤 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
29	6, 28	syl5bi 232	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∃𝑧 ∈ 𝐵 𝐶𝑅𝑧 → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
30	4, 29	impbid 202	1 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574   class class class wbr 4653   Or wor 5034  supcsup 8346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-po 5035  df-so 5036  df-iota 5851  df-riota 6611  df-sup 8348

This theorem is referenced by:  infglbb  8397  suprlub  10987  supxrlub  12155