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Mirrors > Home > NFE Home > Th. List > ssofeq | GIF version |
Description: When A and B are subsets of C, equality depends only on the elements of C. (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
ssofeq | ⊢ ((A ⊆ C ∧ B ⊆ C) → (A = B ↔ ∀x ∈ C (x ∈ A ↔ x ∈ B))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssofss 4076 | . . 3 ⊢ (A ⊆ C → (A ⊆ B ↔ ∀x ∈ C (x ∈ A → x ∈ B))) | |
2 | ssofss 4076 | . . 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 3287 | . 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A)) | |
5 | ralbiim 2751 | . 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))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∀wral 2614 ⊆ wss 3257 |
This theorem was proved from 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 theorem 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 2478 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-ss 3259 |
This theorem is referenced by: eqpw1 4162 pw111 4170 eqrelk 4212 sikexlem 4295 insklem 4304 |
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