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Theorem supubti 6405
Description: A supremum is an upper bound. See also supclti 6404 and suplubti 6406.

This proof demonstrates how to expand an iota-based definition (df-iota 4895) using riotacl2 5509.

(Contributed by Jim Kingdon, 24-Nov-2021.)

Hypotheses
Ref Expression
supmoti.ti ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
supclti.2 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
Assertion
Ref Expression
supubti (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
Distinct variable groups:   𝑢,𝐴,𝑣,𝑥   𝑦,𝐴,𝑥,𝑧   𝑥,𝐵,𝑦,𝑧   𝑢,𝑅,𝑣,𝑥   𝑦,𝑅,𝑧   𝜑,𝑢,𝑣,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐵(𝑣,𝑢)   𝐶(𝑥,𝑦,𝑧,𝑣,𝑢)

Proof of Theorem supubti
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 simpl 106 . . . . 5 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∀𝑦𝐵 ¬ 𝑥𝑅𝑦)
21a1i 9 . . . 4 (𝑥𝐴 → ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∀𝑦𝐵 ¬ 𝑥𝑅𝑦))
32ss2rabi 3050 . . 3 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} ⊆ {𝑥𝐴 ∣ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦}
4 supmoti.ti . . . . 5 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
5 supclti.2 . . . . 5 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
64, 5supval2ti 6401 . . . 4 (𝜑 → sup(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))))
74, 5supeuti 6400 . . . . 5 (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
8 riotacl2 5509 . . . . 5 (∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
97, 8syl 14 . . . 4 (𝜑 → (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
106, 9eqeltrd 2130 . . 3 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
113, 10sseldi 2971 . 2 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦})
12 breq2 3796 . . . . . . 7 (𝑦 = 𝑤 → (𝑥𝑅𝑦𝑥𝑅𝑤))
1312notbid 602 . . . . . 6 (𝑦 = 𝑤 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑤))
1413cbvralv 2550 . . . . 5 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑤𝐵 ¬ 𝑥𝑅𝑤)
15 breq1 3795 . . . . . . 7 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (𝑥𝑅𝑤 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
1615notbid 602 . . . . . 6 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (¬ 𝑥𝑅𝑤 ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
1716ralbidv 2343 . . . . 5 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑤𝐵 ¬ 𝑥𝑅𝑤 ↔ ∀𝑤𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
1814, 17syl5bb 185 . . . 4 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑤𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
1918elrab 2721 . . 3 (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦} ↔ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑤𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2019simprbi 264 . 2 (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦} → ∀𝑤𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤)
21 breq2 3796 . . . 4 (𝑤 = 𝐶 → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
2221notbid 602 . . 3 (𝑤 = 𝐶 → (¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
2322rspccv 2670 . 2 (∀𝑤𝐵 ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
2411, 20, 233syl 17 1 (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wral 2323  wrex 2324  ∃!wreu 2325  {crab 2327   class class class wbr 3792  crio 5495  supcsup 6388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-riota 5496  df-sup 6390
This theorem is referenced by:  supisoti  6414
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