Theorem supmax 9458

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 Jeff Hoffman, 17-Jun-2008.) (Proof shortened by OpenAI, 30-Mar-2020.)

Hypotheses

Ref	Expression
supmax.1	⊢ (𝜑 → 𝑅 Or 𝐴)
supmax.2	⊢ (𝜑 → 𝐶 ∈ 𝐴)
supmax.3	⊢ (𝜑 → 𝐶 ∈ 𝐵)
supmax.4	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦)

Assertion

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

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

Proof of Theorem supmax

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		supmax.1	. 2 ⊢ (𝜑 → 𝑅 Or 𝐴)
2		supmax.2	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
3		supmax.4	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦)
4		supmax.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
5		simprr 772	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → 𝑦𝑅𝐶)
6		breq2 5151	. . . 4 ⊢ (𝑧 = 𝐶 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝐶))
7	6	rspcev 3612	. . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑦𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)
8	4, 5, 7	syl2an2r 684	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)
9	1, 2, 3, 8	eqsupd 9448	1 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∃wrex 3071   class class class wbr 5147   Or wor 5586  supcsup 9431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-po 5587  df-so 5588  df-iota 6492  df-riota 7360  df-sup 9433

This theorem is referenced by:  suppr  9462  gsumesum  32995  supfz  34636  mblfinlem2  36464