Theorem supnub 9420

Description: An upper bound is not less than the supremum. (Contributed by NM, 13-Oct-2004.)

Hypotheses

Ref	Expression
supmo.1	⊢ (𝜑 → 𝑅 Or 𝐴)
supcl.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))

Assertion

Ref	Expression
supnub	⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝐶𝑅𝑧) → ¬ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝑅,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑧,𝐶

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦)

Proof of Theorem supnub

Step	Hyp	Ref	Expression
1		supmo.1	. . . . . 6 ⊢ (𝜑 → 𝑅 Or 𝐴)
2		supcl.2	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
3	1, 2	suplub 9418	. . . . 5 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
4	3	expdimp 452	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
5		dfrex2 3057	. . . 4 ⊢ (∃𝑧 ∈ 𝐵 𝐶𝑅𝑧 ↔ ¬ ∀𝑧 ∈ 𝐵 ¬ 𝐶𝑅𝑧)
6	4, 5	imbitrdi 251	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ¬ ∀𝑧 ∈ 𝐵 ¬ 𝐶𝑅𝑧))
7	6	con2d 134	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∀𝑧 ∈ 𝐵 ¬ 𝐶𝑅𝑧 → ¬ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
8	7	expimpd 453	1 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝐶𝑅𝑧) → ¬ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   class class class wbr 5110   Or wor 5548  supcsup 9398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-po 5549  df-so 5550  df-iota 6467  df-riota 7347  df-sup 9400

This theorem is referenced by:  supssd  9421  dgrlb  26148  supinf  42237