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Mirrors > Home > MPE Home > Th. List > sneqbg | Structured version Visualization version GIF version |
Description: Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.) |
Ref | Expression |
---|---|
sneqbg | ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneqrg 4730 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵)) | |
2 | sneq 4535 | . 2 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
3 | 1, 2 | impbid1 228 | 1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 {csn 4525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-sn 4526 |
This theorem is referenced by: suppval1 7819 suppsnop 7827 fseqdom 9437 infpwfidom 9439 canthwe 10062 s111 13960 initoid 17257 termoid 17258 embedsetcestrclem 17399 mat1dimelbas 21076 mat1dimbas 21077 unidifsnne 30308 altopthg 33541 altopthbg 33542 bj-snglc 34405 f1omptsnlem 34753 fvineqsnf1 34827 extid 35728 eusnsn 43618 |
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