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Mirrors > Home > MPE Home > Th. List > sneqrg | Structured version Visualization version GIF version |
Description: Closed form of sneqr 4763. (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 4589 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
2 | eleq2 2898 | . . 3 ⊢ ({𝐴} = {𝐵} → (𝐴 ∈ {𝐴} ↔ 𝐴 ∈ {𝐵})) | |
3 | 1, 2 | syl5ibcom 246 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 ∈ {𝐵})) |
4 | elsng 4571 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) | |
5 | 3, 4 | sylibd 240 | 1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {csn 4557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-sn 4558 |
This theorem is referenced by: sneqr 4763 sneqbg 4766 preimane 30343 altopth1 33323 altopth2 33324 |
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