Theorem clss2lem 44151

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 494	. . . 4 ⊢ (𝜑 → ((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓))
3	2	alrimiv 1946	. . 3 ⊢ (𝜑 → ∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓))
4		pm5.3 580	. . . . 5 ⊢ (((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓) ↔ ((𝑋 ⊆ 𝑥 ∧ 𝜒) → (𝑋 ⊆ 𝑥 ∧ 𝜓)))
5	4	albii 1838	. . . 4 ⊢ (∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓) ↔ ∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → (𝑋 ⊆ 𝑥 ∧ 𝜓)))
6		ss2ab 4014	. . . 4 ⊢ ({𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)} ⊆ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ↔ ∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → (𝑋 ⊆ 𝑥 ∧ 𝜓)))
7	5, 6	bitr4i 280	. . 3 ⊢ (∀𝑥((𝑋 ⊆ 𝑥 ∧ 𝜒) → 𝜓) ↔ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)} ⊆ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)})
8	3, 7	sylib 220	. 2 ⊢ (𝜑 → {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)} ⊆ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)})
9		intss 4926	. 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 399  ∀wal 1557  {cab 2739   ⊆ wss 3904  ∩ cint 4904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-ss 3921  df-int 4905

This theorem is referenced by: (None)