Theorem disjeq1 3726

Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjeq1	⊢ (A = B → (Disj x ∈ A 𝐶 ↔ Disj x ∈ B 𝐶))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   𝐶(x)

Proof of Theorem disjeq1

Step	Hyp	Ref	Expression
1		eqimss2 2975	. . 3 ⊢ (A = B → B ⊆ A)
2		disjss1 3725	. . 3 ⊢ (B ⊆ A → (Disj x ∈ A 𝐶 → Disj x ∈ B 𝐶))
3	1, 2	syl 14	. 2 ⊢ (A = B → (Disj x ∈ A 𝐶 → Disj x ∈ B 𝐶))
4		eqimss 2974	. . 3 ⊢ (A = B → A ⊆ B)
5		disjss1 3725	. . 3 ⊢ (A ⊆ B → (Disj x ∈ B 𝐶 → Disj x ∈ A 𝐶))
6	4, 5	syl 14	. 2 ⊢ (A = B → (Disj x ∈ B 𝐶 → Disj x ∈ A 𝐶))
7	3, 6	impbid 120	1 ⊢ (A = B → (Disj x ∈ A 𝐶 ↔ Disj x ∈ B 𝐶))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1228   ⊆ wss 2894  Disj wdisj 3719

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-rmo 2292  df-in 2901  df-ss 2908  df-disj 3720

This theorem is referenced by:  disjeq1d  3727