New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > eqrelkriiv | GIF version |
Description: Equality for two Kuratowski relationships. (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
eqrelkriiv.1 | ⊢ A ⊆ (V ×k V) |
eqrelkriiv.2 | ⊢ B ⊆ (V ×k V) |
eqrelkriiv.3 | ⊢ (⟪x, y⟫ ∈ A ↔ ⟪x, y⟫ ∈ B) |
Ref | Expression |
---|---|
eqrelkriiv | ⊢ A = B |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrelkriiv.3 | . . 3 ⊢ (⟪x, y⟫ ∈ A ↔ ⟪x, y⟫ ∈ B) | |
2 | 1 | gen2 1547 | . 2 ⊢ ∀x∀y(⟪x, y⟫ ∈ A ↔ ⟪x, y⟫ ∈ B) |
3 | eqrelkriiv.1 | . . 3 ⊢ A ⊆ (V ×k V) | |
4 | eqrelkriiv.2 | . . 3 ⊢ B ⊆ (V ×k V) | |
5 | eqrelk 4212 | . . 3 ⊢ ((A ⊆ (V ×k V) ∧ B ⊆ (V ×k V)) → (A = B ↔ ∀x∀y(⟪x, y⟫ ∈ A ↔ ⟪x, y⟫ ∈ B))) | |
6 | 3, 4, 5 | mp2an 653 | . 2 ⊢ (A = B ↔ ∀x∀y(⟪x, y⟫ ∈ A ↔ ⟪x, y⟫ ∈ B)) |
7 | 2, 6 | mpbir 200 | 1 ⊢ A = B |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∀wal 1540 = wceq 1642 ∈ wcel 1710 Vcvv 2859 ⊆ wss 3257 ⟪copk 4057 ×k cxpk 4174 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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 ax-nin 4078 ax-sn 4087 |
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-ne 2518 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-pr 3742 df-opk 4058 df-xpk 4185 |
This theorem is referenced by: cnvkxpk 4276 inxpk 4277 dfidk2 4313 |
Copyright terms: Public domain | W3C validator |