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Mirrors > Home > ILE Home > Th. List > eqrd | GIF version |
Description: Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.) |
Ref | Expression |
---|---|
eqrd.0 | ⊢ Ⅎ𝑥𝜑 |
eqrd.1 | ⊢ Ⅎ𝑥𝐴 |
eqrd.2 | ⊢ Ⅎ𝑥𝐵 |
eqrd.3 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Ref | Expression |
---|---|
eqrd | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrd.0 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | eqrd.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | eqrd.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | eqrd.3 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
5 | 4 | biimpd 132 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
6 | 1, 2, 3, 5 | ssrd 2950 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
7 | 4 | biimprd 147 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)) |
8 | 1, 3, 2, 7 | ssrd 2950 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
9 | 6, 8 | eqssd 2962 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 Ⅎwnf 1349 ∈ wcel 1393 Ⅎwnfc 2165 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-in 2924 df-ss 2931 |
This theorem is referenced by: (None) |
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