Theorem disjss2 5075

Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjss2	⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵))

Proof of Theorem disjss2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3906	. . . . 5 ⊢ (𝐵 ⊆ 𝐶 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
2	1	ralimi 3102	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
3		rmoim 3700	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) → (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
4	2, 3	syl 18	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
5	4	alimdv 1950	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
6		df-disj 5073	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
7		df-disj 5073	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
8	5, 6, 7	3imtr4g 299	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567   ∈ wcel 2146  ∀wral 3079  ∃*wrmo 3367   ⊆ wss 3899  Disj wdisj 5072

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-mo 2567  df-clel 2838  df-ral 3080  df-rmo 3368  df-ss 3900  df-disj 5073

This theorem is used by:  disjeq2  5076  0disj  5098  uniioombllem2  25916  uniioombllem4  25919  disjxwwlksn  30505  disjxwwlkn  30514  fusgreghash2wspv  30948  fsumiunss  46726