Theorem 0elsros 34178

Description: A semiring of sets contains the empty set. (Contributed by Thierry Arnoux, 18-Jul-2020.)

Hypothesis

Ref	Expression
issros.1	⊢ 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∩ 𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))}

Assertion

Ref	Expression
0elsros	⊢ (𝑆 ∈ 𝑁 → ∅ ∈ 𝑆)

Distinct variable groups:   𝑡,𝑠,𝑥,𝑦   𝑂,𝑠   𝑆,𝑠,𝑥,𝑦,𝑧

Allowed substitution hints:   𝑆(𝑡)   𝑁(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑧,𝑡)

Proof of Theorem 0elsros

Step	Hyp	Ref	Expression
1		issros.1	. . 3 ⊢ 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∩ 𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))}
2	1	issros 34176	. 2 ⊢ (𝑆 ∈ 𝑁 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∩ 𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧))))
3	2	simp2bi 1147	1 ⊢ (𝑆 ∈ 𝑁 → ∅ ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  {crab 3436   ∖ cdif 3948   ∩ cin 3950  ∅c0 4333  𝒫 cpw 4600  ∪ cuni 4907  Disj wdisj 5110  Fincfn 8985

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-ss 3968  df-pw 4602

This theorem is referenced by: (None)