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Mirrors > Home > MPE Home > Th. List > 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 1805 | . . 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 3986 | . 2 ⊢ (⊤ → 𝐴 = 𝐵) |
7 | 6 | mptru 1544 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 Ⅎwnfc 2961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-cleq 2814 df-clel 2893 df-nfc 2963 |
This theorem is referenced by: rnep 5797 difrab2 30261 esum2dlem 31351 eulerpartlemn 31639 |
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