Theorem clss2lem 42938

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 1922	. . 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 1813	. . . 4 ⊢ (∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓) ↔ ∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → (𝑋 ⊆ 𝑥 ∧ 𝜓)))
6		ss2ab 4051	. . . 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 217	. 2 ⊢ (𝜑 → {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)} ⊆ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)})
9		intss 4966	. 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 1531  {cab 2703   ⊆ wss 3943  ∩ cint 4943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-v 3470  df-in 3950  df-ss 3960  df-int 4944

This theorem is referenced by: (None)