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Theorem suplub2 8636
Description: Bidirectional form of suplub 8635. (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 8635 . . 3 (𝜑 → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))
43expdimp 446 . 2 ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧))
5 breq2 4877 . . . 4 (𝑧 = 𝑤 → (𝐶𝑅𝑧𝐶𝑅𝑤))
65cbvrexv 3384 . . 3 (∃𝑧𝐵 𝐶𝑅𝑧 ↔ ∃𝑤𝐵 𝐶𝑅𝑤)
7 breq2 4877 . . . . . . 7 (sup(𝐵, 𝐴, 𝑅) = 𝑤 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅𝑤))
87biimprd 240 . . . . . 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 717 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → 𝑅 Or 𝐴)
11 simplr 785 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → 𝐶𝐴)
12 suplub2.3 . . . . . . . . 9 (𝜑𝐵𝐴)
1312adantr 474 . . . . . . . 8 ((𝜑𝐶𝐴) → 𝐵𝐴)
1413sselda 3827 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → 𝑤𝐴)
151, 2supcl 8633 . . . . . . . 8 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
1615ad2antrr 717 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
17 sotr 5285 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝑤𝐴 ∧ sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)) → ((𝐶𝑅𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)) → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
1810, 11, 14, 16, 17syl13anc 1495 . . . . . 6 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → ((𝐶𝑅𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)) → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
1918expcomd 408 . . . . 5 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → (𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅))))
201, 2supub 8634 . . . . . . . 8 (𝜑 → (𝑤𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2120adantr 474 . . . . . . 7 ((𝜑𝐶𝐴) → (𝑤𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2221imp 397 . . . . . 6 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤)
23 sotric 5289 . . . . . . . 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 870 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ (sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅))))
2524con2bid 346 . . . . . 6 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → ((sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)) ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2622, 25mpbird 249 . . . . 5 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
279, 19, 26mpjaod 891 . . . 4 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
2827rexlimdva 3240 . . 3 ((𝜑𝐶𝐴) → (∃𝑤𝐵 𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
296, 28syl5bi 234 . 2 ((𝜑𝐶𝐴) → (∃𝑧𝐵 𝐶𝑅𝑧𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
304, 29impbid 204 1 ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧𝐵 𝐶𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 878   = wceq 1656  wcel 2164  wral 3117  wrex 3118  wss 3798   class class class wbr 4873   Or wor 5262  supcsup 8615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-po 5263  df-so 5264  df-iota 6086  df-riota 6866  df-sup 8617
This theorem is referenced by:  infglbb  8666  suprlub  11317  supxrlub  12443
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