Theorem supmax 8373

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 796	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → 𝑦𝑅𝐶)
6		breq2 4657	. . . 4 ⊢ (𝑧 = 𝐶 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝐶))
7	6	rspcev 3309	. . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑦𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)
8	4, 5, 7	syl2an2r 876	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)
9	1, 2, 3, 8	eqsupd 8363	1 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913   class class class wbr 4653   Or wor 5034  supcsup 8346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-po 5035  df-so 5036  df-iota 5851  df-riota 6611  df-sup 8348

This theorem is referenced by:  suppr  8377  gsumesum  30121  supfz  31613  inffzOLD  31615  mblfinlem2  33447