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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 1799 | . . 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 3997 | . 2 ⊢ (⊤ → 𝐴 = 𝐵) |
7 | 6 | mptru 1541 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 Ⅎwnfc 2879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-cleq 2720 df-clel 2806 df-nfc 2881 |
This theorem is referenced by: rnep 5923 difrab2 32289 esum2dlem 33705 eulerpartlemn 33995 |
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