Theorem supclti 7112

Description: A supremum belongs to its base class (closure law). See also supubti 7113 and suplubti 7114. (Contributed by Jim Kingdon, 24-Nov-2021.)

Hypotheses

Ref	Expression
supmoti.ti	⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
supclti.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))

Assertion

Ref	Expression
supclti	⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)

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)

Proof of Theorem supclti

Step	Hyp	Ref	Expression
1		supmoti.ti	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
2		supclti.2	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
3	1, 2	supval2ti 7109	. 2 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
4	1, 2	supeuti 7108	. . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
5		riotacl 5924	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ∈ 𝐴)
6	4, 5	syl 14	. 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ∈ 𝐴)
7	3, 6	eqeltrd 2283	1 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  ∃!wreu 2487   class class class wbr 4048  ℩crio 5908  supcsup 7096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3001  df-un 3172  df-in 3174  df-ss 3181  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-br 4049  df-iota 5238  df-riota 5909  df-sup 7098

This theorem is referenced by:  suplub2ti  7115  supelti  7116  supisoti  7124  infclti  7137  inflbti  7138  infglbti  7139  suprubex  9037  suprleubex  9040  sup3exmid  9043  suprzclex  9484  supminfex  9731  zsupcl  10387  maxleast  11574  dvdslegcd  12335