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Theorem suplub2 8250
 Description: Bidirectional form of suplub 8249. (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 8249 . . 3 (𝜑 → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))
43expdimp 452 . 2 ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧))
5 breq2 4587 . . . 4 (𝑧 = 𝑤 → (𝐶𝑅𝑧𝐶𝑅𝑤))
65cbvrexv 3148 . . 3 (∃𝑧𝐵 𝐶𝑅𝑧 ↔ ∃𝑤𝐵 𝐶𝑅𝑤)
7 breq2 4587 . . . . . . 7 (sup(𝐵, 𝐴, 𝑅) = 𝑤 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅𝑤))
87biimprd 237 . . . . . 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 758 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → 𝑅 Or 𝐴)
11 simplr 788 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → 𝐶𝐴)
12 suplub2.3 . . . . . . . . 9 (𝜑𝐵𝐴)
1312adantr 480 . . . . . . . 8 ((𝜑𝐶𝐴) → 𝐵𝐴)
1413sselda 3568 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → 𝑤𝐴)
151, 2supcl 8247 . . . . . . . 8 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
1615ad2antrr 758 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
17 sotr 4981 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝑤𝐴 ∧ sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)) → ((𝐶𝑅𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)) → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
1810, 11, 14, 16, 17syl13anc 1320 . . . . . 6 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → ((𝐶𝑅𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)) → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
1918expcomd 453 . . . . 5 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → (𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅))))
201, 2supub 8248 . . . . . . . 8 (𝜑 → (𝑤𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2120adantr 480 . . . . . . 7 ((𝜑𝐶𝐴) → (𝑤𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2221imp 444 . . . . . 6 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤)
23 sotric 4985 . . . . . . . 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 1316 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ (sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅))))
2524con2bid 343 . . . . . 6 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → ((sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)) ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2622, 25mpbird 246 . . . . 5 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
279, 19, 26mpjaod 395 . . . 4 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
2827rexlimdva 3013 . . 3 ((𝜑𝐶𝐴) → (∃𝑤𝐵 𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
296, 28syl5bi 231 . 2 ((𝜑𝐶𝐴) → (∃𝑧𝐵 𝐶𝑅𝑧𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
304, 29impbid 201 1 ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧𝐵 𝐶𝑅𝑧))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   class class class wbr 4583   Or wor 4958  supcsup 8229 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-po 4959  df-so 4960  df-iota 5768  df-riota 6511  df-sup 8231 This theorem is referenced by:  infglbb  8280  suprlub  10864  supxrlub  12027
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