Theorem clss2lem 43624

Description: The closure of a property is a superset of the closure of a less restrictive property. (Contributed by RP, 24-Jul-2020.)

Hypothesis

Ref	Expression
clss2lem.1	⊢ (𝜑 → (𝜒 → 𝜓))

Assertion

Ref	Expression
clss2lem	⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)})

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝑋(𝑥)

Proof of Theorem clss2lem

Step	Hyp	Ref	Expression
1		clss2lem.1	. . . . 5 ⊢ (𝜑 → (𝜒 → 𝜓))
2	1	adantld 490	. . . 4 ⊢ (𝜑 → ((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓))
3	2	alrimiv 1927	. . 3 ⊢ (𝜑 → ∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓))
4		pm5.3 572	. . . . 5 ⊢ (((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓) ↔ ((𝑋 ⊆ 𝑥 ∧ 𝜒) → (𝑋 ⊆ 𝑥 ∧ 𝜓)))
5	4	albii 1819	. . . 4 ⊢ (∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓) ↔ ∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → (𝑋 ⊆ 𝑥 ∧ 𝜓)))
6		ss2ab 4062	. . . 4 ⊢ ({𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)} ⊆ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ↔ ∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → (𝑋 ⊆ 𝑥 ∧ 𝜓)))
7	5, 6	bitr4i 278	. . 3 ⊢ (∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓) ↔ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)} ⊆ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)})
8	3, 7	sylib 218	. 2 ⊢ (𝜑 → {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)} ⊆ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)})
9		intss 4969	. 2 ⊢ ({𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)} ⊆ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)})
10	8, 9	syl 17	1 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)})

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538  {cab 2714   ⊆ wss 3951  ∩ cint 4946

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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-ss 3968  df-int 4947

This theorem is referenced by: (None)