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Theorem suplub2 8924
Description: Bidirectional form of suplub 8923. (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 8923 . . 3 (𝜑 → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))
43expdimp 455 . 2 ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧))
5 breq2 5069 . . . 4 (𝑧 = 𝑤 → (𝐶𝑅𝑧𝐶𝑅𝑤))
65cbvrexvw 3450 . . 3 (∃𝑧𝐵 𝐶𝑅𝑧 ↔ ∃𝑤𝐵 𝐶𝑅𝑤)
7 breq2 5069 . . . . . . 7 (sup(𝐵, 𝐴, 𝑅) = 𝑤 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅𝑤))
87biimprd 250 . . . . . 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 483 . . . . . . . 8 ((𝜑𝐶𝐴) → 𝐵𝐴)
1413sselda 3966 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → 𝑤𝐴)
151, 2supcl 8921 . . . . . . . 8 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
1615ad2antrr 724 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
17 sotr 5496 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝑤𝐴 ∧ sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)) → ((𝐶𝑅𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)) → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
1810, 11, 14, 16, 17syl13anc 1368 . . . . . 6 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → ((𝐶𝑅𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)) → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
1918expcomd 419 . . . . 5 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → (𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅))))
201, 2supub 8922 . . . . . . . 8 (𝜑 → (𝑤𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2120adantr 483 . . . . . . 7 ((𝜑𝐶𝐴) → (𝑤𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2221imp 409 . . . . . 6 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤)
23 sotric 5500 . . . . . . . 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 834 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ (sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅))))
2524con2bid 357 . . . . . 6 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → ((sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)) ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2622, 25mpbird 259 . . . . 5 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
279, 19, 26mpjaod 856 . . . 4 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
2827rexlimdva 3284 . . 3 ((𝜑𝐶𝐴) → (∃𝑤𝐵 𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
296, 28syl5bi 244 . 2 ((𝜑𝐶𝐴) → (∃𝑧𝐵 𝐶𝑅𝑧𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
304, 29impbid 214 1 ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧𝐵 𝐶𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843   = wceq 1533  wcel 2110  wral 3138  wrex 3139  wss 3935   class class class wbr 5065   Or wor 5472  supcsup 8903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-po 5473  df-so 5474  df-iota 6313  df-riota 7113  df-sup 8905
This theorem is referenced by:  infglbb  8954  suprlub  11604  supxrlub  12717
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