![]() |
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 3635 | . . 3 ⊢ 𝐴 ∈ {𝐴} |
3 | eleq2 2251 | . . 3 ⊢ ({𝐴} = {𝐵} → (𝐴 ∈ {𝐴} ↔ 𝐴 ∈ {𝐵})) | |
4 | 2, 3 | mpbii 148 | . 2 ⊢ ({𝐴} = {𝐵} → 𝐴 ∈ {𝐵}) |
5 | 1 | elsn 3620 | . 2 ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵) |
6 | 4, 5 | sylib 122 | 1 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 Vcvv 2749 {csn 3604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-sn 3610 |
This theorem is referenced by: sneqrg 3774 opth1 4248 cc2lem 7279 |
Copyright terms: Public domain | W3C validator |