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Mirrors > Home > ILE Home > Th. List > sneqi | GIF version |
Description: Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
Ref | Expression |
---|---|
sneqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
sneqi | ⊢ {𝐴} = {𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | sneq 3592 | . 2 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ {𝐴} = {𝐵} |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 {csn 3581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-sn 3587 |
This theorem is referenced by: fnressn 5679 fressnfv 5680 snriota 5835 xpassen 6804 ennnfonelem1 12349 strle1g 12495 |
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