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Mirrors > Home > MPE Home > Th. List > sneqrg | Structured version Visualization version GIF version |
Description: Closed form of sneqr 4311. (Contributed by Scott Fenton, 1-Apr-2011.) (Proof shortened by JJ, 23-Jul-2021.) |
Ref | Expression |
---|---|
sneqrg | ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snidg 4153 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
2 | eleq2 2677 | . . 3 ⊢ ({𝐴} = {𝐵} → (𝐴 ∈ {𝐴} ↔ 𝐴 ∈ {𝐵})) | |
3 | 1, 2 | syl5ibcom 234 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 ∈ {𝐵})) |
4 | elsng 4139 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) | |
5 | 3, 4 | sylibd 228 | 1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sn 4126 |
This theorem is referenced by: sneqr 4311 sneqbg 4314 altopth1 31242 altopth2 31243 |
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