Theorem disjss1f 32990

Description: A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017.)

Hypotheses

Ref	Expression
disjss1f.1	⊢ Ⅎ𝑥𝐴
disjss1f.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
disjss1f	⊢ (𝐴 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶))

Proof of Theorem disjss1f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjss1f.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2		disjss1f.2	. . . 4 ⊢ Ⅎ𝑥𝐵
3	1, 2	ssrmof 4006	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
4	3	alimdv 1949	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
5		df-disj 5079	. 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)
6		df-disj 5079	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
7	4, 5, 6	3imtr4g 299	1 ⊢ (𝐴 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   ∈ wcel 2146  Ⅎwnfc 2912  ∃*wrmo 3370   ⊆ wss 3906  Disj wdisj 5078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-mo 2569  df-clel 2840  df-nfc 2914  df-rmo 3371  df-ss 3923  df-disj 5079

This theorem is used by:  disjeq1f  32991  esumrnmpt2  34524  measvuni  34671