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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 4772 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵)) | |
2 | sneq 4579 | . 2 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
3 | 1, 2 | impbid1 227 | 1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 {csn 4569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-sn 4570 |
This theorem is referenced by: suppval1 7838 suppsnop 7846 fseqdom 9454 infpwfidom 9456 canthwe 10075 s111 13971 initoid 17267 termoid 17268 embedsetcestrclem 17409 mat1dimelbas 21082 mat1dimbas 21083 unidifsnne 30298 altopthg 33430 altopthbg 33431 bj-snglc 34283 f1omptsnlem 34619 fvineqsnf1 34693 extid 35570 eusnsn 43268 |
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