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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 3481 | . . 3 ⊢ 𝐴 ∈ {𝐴} |
3 | eleq2 2152 | . . 3 ⊢ ({𝐴} = {𝐵} → (𝐴 ∈ {𝐴} ↔ 𝐴 ∈ {𝐵})) | |
4 | 2, 3 | mpbii 147 | . 2 ⊢ ({𝐴} = {𝐵} → 𝐴 ∈ {𝐵}) |
5 | 1 | elsn 3468 | . 2 ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵) |
6 | 4, 5 | sylib 121 | 1 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1290 ∈ wcel 1439 Vcvv 2622 {csn 3452 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2624 df-sn 3458 |
This theorem is referenced by: sneqrg 3614 opth1 4074 |
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