Theorem supmax 8540

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 813	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → 𝑦𝑅𝐶)
6		breq2 4808	. . . 4 ⊢ (𝑧 = 𝐶 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝐶))
7	6	rspcev 3449	. . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑦𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)
8	4, 5, 7	syl2an2r 911	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)
9	1, 2, 3, 8	eqsupd 8530	1 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∃wrex 3051   class class class wbr 4804   Or wor 5186  supcsup 8513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-po 5187  df-so 5188  df-iota 6012  df-riota 6775  df-sup 8515

This theorem is referenced by:  suppr  8544  gsumesum  30451  supfz  31941  inffzOLD  31943  mblfinlem2  33778