Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sneqr | GIF version |
Description: If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.) |
Ref | Expression |
---|---|
sneqr.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
sneqr | ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneqr.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | 1 | snid 3402 | . . 3 ⊢ 𝐴 ∈ {𝐴} |
3 | eleq2 2101 | . . 3 ⊢ ({𝐴} = {𝐵} → (𝐴 ∈ {𝐴} ↔ 𝐴 ∈ {𝐵})) | |
4 | 2, 3 | mpbii 136 | . 2 ⊢ ({𝐴} = {𝐵} → 𝐴 ∈ {𝐵}) |
5 | 1 | elsn 3391 | . 2 ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵) |
6 | 4, 5 | sylib 127 | 1 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 Vcvv 2557 {csn 3375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-sn 3381 |
This theorem is referenced by: sneqrg 3533 opth1 3973 |
Copyright terms: Public domain | W3C validator |