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| Mirrors > Home > MPE Home > Th. List > sneq | Structured version Visualization version GIF version | ||
| Description: Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.) |
| Ref | Expression |
|---|---|
| sneq | ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqeq2 2749 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑥 = 𝐴 ↔ 𝑥 = 𝐵)) | |
| 2 | 1 | abbidv 2808 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝐵}) |
| 3 | df-sn 4627 | . 2 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴} | |
| 4 | df-sn 4627 | . 2 ⊢ {𝐵} = {𝑥 ∣ 𝑥 = 𝐵} | |
| 5 | 2, 3, 4 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) |
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