Theorem ssofeq 4078

Description: When A and B are subsets of C, equality depends only on the elements of C. (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
ssofeq	⊢ ((A ⊆ C ∧ B ⊆ C) → (A = B ↔ ∀x ∈ C (x ∈ A ↔ x ∈ B)))

Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem ssofeq

Step	Hyp	Ref	Expression
1		ssofss 4077	. . 3 ⊢ (A ⊆ C → (A ⊆ B ↔ ∀x ∈ C (x ∈ A → x ∈ B)))
2		ssofss 4077	. . 3 ⊢ (B ⊆ C → (B ⊆ A ↔ ∀x ∈ C (x ∈ B → x ∈ A)))
3	1, 2	bi2anan9 843	. 2 ⊢ ((A ⊆ C ∧ B ⊆ C) → ((A ⊆ B ∧ B ⊆ A) ↔ (∀x ∈ C (x ∈ A → x ∈ B) ∧ ∀x ∈ C (x ∈ B → x ∈ A))))
4		eqss 3288	. 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
5		ralbiim 2752	. 2 ⊢ (∀x ∈ C (x ∈ A ↔ x ∈ B) ↔ (∀x ∈ C (x ∈ A → x ∈ B) ∧ ∀x ∈ C (x ∈ B → x ∈ A)))
6	3, 4, 5	3bitr4g 279	1 ⊢ ((A ⊆ C ∧ B ⊆ C) → (A = B ↔ ∀x ∈ C (x ∈ A ↔ x ∈ B)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615   ⊆ wss 3258

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-ss 3260

This theorem is used by:  eqpw1  4163  pw111  4171  eqrelk  4213  sikexlem  4296  insklem  4305