Theorem disjss1 5074

Description: A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjss1	⊢ (𝐴 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem disjss1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3931	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	anim1d 620	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
3	2	moimdv 2574	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
4	3	alimdv 1937	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)))
5		dfdisj2 5070	. 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
6		dfdisj2 5070	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶))
7	4, 5, 6	3imtr4g 298	1 ⊢ (𝐴 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1559   ∈ wcel 2143  ∃*wmo 2565   ⊆ wss 3905  Disj wdisj 5068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-mo 2567  df-clel 2838  df-rmo 3368  df-ss 3922  df-disj 5069

This theorem is referenced by:  disjeq1  5075  disjx0  5096  disjxiun  5098  disjss3  5100  volfiniun  25610  uniioovol  25642  uniioombllem4  25649  disjiunel  32797  tocyccntz  33325  carsggect  34616  carsgclctunlem2  34617  omsmeas  34621  sibfof  34638  disjf1o  45770  fsumiunss  46152  sge0iunmptlemre  46990  meadjiunlem  47040  meaiuninclem  47055