Theorem ssdif 3262

Description: Difference law for subsets. (Contributed by NM, 28-May-1998.)

Assertion

Ref	Expression
ssdif	⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))

Proof of Theorem ssdif

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3141	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	anim1d 334	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
3		eldif 3130	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶))
4		eldif 3130	. . 3 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
5	2, 3, 4	3imtr4g 204	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ (𝐴 ∖ 𝐶) → 𝑥 ∈ (𝐵 ∖ 𝐶)))
6	5	ssrdv 3153	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∈ wcel 2141   ∖ cdif 3118   ⊆ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134

This theorem is referenced by:  ssdifd  3263  phpm  6843  difinfinf  7078