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Theorem suplub2 9491
Description: Bidirectional form of suplub 9490. (Contributed by Mario Carneiro, 6-Sep-2014.)
Hypotheses
Ref Expression
supmo.1 (𝜑𝑅 Or 𝐴)
supcl.2 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
suplub2.3 (𝜑𝐵𝐴)
Assertion
Ref Expression
suplub2 ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧𝐵 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.
StepHypRef Expression
1 supmo.1 . . . 4 (𝜑𝑅 Or 𝐴)
2 supcl.2 . . . 4 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
31, 2suplub 9490 . . 3 (𝜑 → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))
43expdimp 451 . 2 ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧))
5 breq2 5153 . . . 4 (𝑧 = 𝑤 → (𝐶𝑅𝑧𝐶𝑅𝑤))
65cbvrexvw 3225 . . 3 (∃𝑧𝐵 𝐶𝑅𝑧 ↔ ∃𝑤𝐵 𝐶𝑅𝑤)
7 breq2 5153 . . . . . . 7 (sup(𝐵, 𝐴, 𝑅) = 𝑤 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅𝑤))
87biimprd 247 . . . . . 6 (sup(𝐵, 𝐴, 𝑅) = 𝑤 → (𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
98a1i 11 . . . . 5 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (sup(𝐵, 𝐴, 𝑅) = 𝑤 → (𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅))))
101ad2antrr 724 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → 𝑅 Or 𝐴)
11 simplr 767 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → 𝐶𝐴)
12 suplub2.3 . . . . . . . . 9 (𝜑𝐵𝐴)
1312adantr 479 . . . . . . . 8 ((𝜑𝐶𝐴) → 𝐵𝐴)
1413sselda 3976 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → 𝑤𝐴)
151, 2supcl 9488 . . . . . . . 8 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
1615ad2antrr 724 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
17 sotr 5614 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝑤𝐴 ∧ sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)) → ((𝐶𝑅𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)) → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
1810, 11, 14, 16, 17syl13anc 1369 . . . . . 6 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → ((𝐶𝑅𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)) → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
1918expcomd 415 . . . . 5 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → (𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅))))
201, 2supub 9489 . . . . . . . 8 (𝜑 → (𝑤𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2120adantr 479 . . . . . . 7 ((𝜑𝐶𝐴) → (𝑤𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2221imp 405 . . . . . 6 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤)
23 sotric 5618 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐴𝑤𝐴)) → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ (sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅))))
2410, 16, 14, 23syl12anc 835 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ (sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅))))
2524con2bid 353 . . . . . 6 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → ((sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)) ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2622, 25mpbird 256 . . . . 5 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
279, 19, 26mpjaod 858 . . . 4 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
2827rexlimdva 3144 . . 3 ((𝜑𝐶𝐴) → (∃𝑤𝐵 𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
296, 28biimtrid 241 . 2 ((𝜑𝐶𝐴) → (∃𝑧𝐵 𝐶𝑅𝑧𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
304, 29impbid 211 1 ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧𝐵 𝐶𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845   = wceq 1533  wcel 2098  wral 3050  wrex 3059  wss 3944   class class class wbr 5149   Or wor 5589  supcsup 9470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-po 5590  df-so 5591  df-iota 6501  df-riota 7375  df-sup 9472
This theorem is referenced by:  infglbb  9521  suprlub  12216  supxrlub  13344
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