Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqri | Structured version Visualization version GIF version |
Description: Infer equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 7-Oct-2017.) |
Ref | Expression |
---|---|
eqri.1 | ⊢ Ⅎ𝑥𝐴 |
eqri.2 | ⊢ Ⅎ𝑥𝐵 |
eqri.3 | ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) |
Ref | Expression |
---|---|
eqri | ⊢ 𝐴 = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1730 | . . 3 ⊢ Ⅎ𝑥⊤ | |
2 | eqri.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | eqri.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | eqri.3 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
6 | 1, 2, 3, 5 | eqrd 3622 | . 2 ⊢ (⊤ → 𝐴 = 𝐵) |
7 | 6 | trud 1493 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 Ⅎwnfc 2751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-cleq 2615 df-clel 2618 df-nfc 2753 |
This theorem is referenced by: difrab2 29339 esum2dlem 30154 eulerpartlemn 30443 |
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