Theorem supmaxti 7063

Description: The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jim Kingdon, 24-Nov-2021.)

Hypotheses

Ref	Expression
supmaxti.ti	⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
supmaxti.2	⊢ (𝜑 → 𝐶 ∈ 𝐴)
supmaxti.3	⊢ (𝜑 → 𝐶 ∈ 𝐵)
supmaxti.4	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦)

Assertion

Ref	Expression
supmaxti	⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)

Distinct variable groups:   𝑢,𝐴,𝑣,𝑦   𝑢,𝐵,𝑣,𝑦   𝑢,𝐶,𝑣,𝑦   𝑢,𝑅,𝑣,𝑦   𝜑,𝑢,𝑣,𝑦

Proof of Theorem supmaxti

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		supmaxti.ti	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
2		supmaxti.2	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
3		supmaxti.4	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦)
4		supmaxti.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
5		simprr 531	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → 𝑦𝑅𝐶)
6		breq2 4033	. . . 4 ⊢ (𝑥 = 𝐶 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐶))
7	6	rspcev 2864	. . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑦𝑅𝐶) → ∃𝑥 ∈ 𝐵 𝑦𝑅𝑥)
8	4, 5, 7	syl2an2r 595	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑥 ∈ 𝐵 𝑦𝑅𝑥)
9	1, 2, 3, 8	eqsuptid 7056	1 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∃wrex 2473   class class class wbr 4029  supcsup 7041

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-riota 5873  df-sup 7043

This theorem is referenced by:  supsnti  7064  sup3exmid  8976  maxleim  11349  xrmaxleim  11387  supfz  15561