Theorem disjss1f 30335

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 3980	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
4	3	alimdv 1917	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
5		df-disj 4996	. 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)
6		df-disj 4996	. 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))

Syntax hints:   → wi 4  ∀wal 1536   ∈ wcel 2111  Ⅎwnfc 2936  ∃*wrmo 3109   ⊆ wss 3881  Disj wdisj 4995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rmo 3114  df-v 3443  df-in 3888  df-ss 3898  df-disj 4996

This theorem is referenced by:  disjeq1f  30336  esumrnmpt2  31437  measvuni  31583