Theorem sup0riota 9534

Description: The supremum of an empty set is the smallest element of the base set. (Contributed by AV, 4-Sep-2020.)

Assertion

Ref	Expression
sup0riota	⊢ (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦

Proof of Theorem sup0riota

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑅 Or 𝐴 → 𝑅 Or 𝐴)
2	1	supval2 9524	. 2 ⊢ (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧))))
3		ral0 4536	. . . . . 6 ⊢ ∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦
4	3	biantrur 530	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)))
5		rex0 4383	. . . . . . 7 ⊢ ¬ ∃𝑧 ∈ ∅ 𝑦𝑅𝑧
6		imnot 365	. . . . . . 7 ⊢ (¬ ∃𝑧 ∈ ∅ 𝑦𝑅𝑧 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ¬ 𝑦𝑅𝑥))
7	5, 6	ax-mp 5	. . . . . 6 ⊢ ((𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ¬ 𝑦𝑅𝑥)
8	7	ralbii 3099	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧) ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)
9	4, 8	bitr3i 277	. . . 4 ⊢ ((∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)) ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)
10	9	a1i 11	. . 3 ⊢ (𝑅 Or 𝐴 → ((∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧)) ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥))
11	10	riotabidv 7406	. 2 ⊢ (𝑅 Or 𝐴 → (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ ∅ ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ ∅ 𝑦𝑅𝑧))) = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥))
12	2, 11	eqtrd 2780	1 ⊢ (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∀wral 3067  ∃wrex 3076  ∅c0 4352   class class class wbr 5166   Or wor 5606  ℩crio 7403  supcsup 9509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-po 5607  df-so 5608  df-iota 6525  df-riota 7404  df-sup 9511

This theorem is referenced by:  sup0  9535