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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 4763 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵)) | |
2 | sneq 4570 | . 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 1533 ∈ wcel 2110 {csn 4560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-sn 4561 |
This theorem is referenced by: suppval1 7830 suppsnop 7838 fseqdom 9446 infpwfidom 9448 canthwe 10067 s111 13963 initoid 17259 termoid 17260 embedsetcestrclem 17401 mat1dimelbas 21074 mat1dimbas 21075 unidifsnne 30290 altopthg 33423 altopthbg 33424 bj-snglc 34276 f1omptsnlem 34611 fvineqsnf1 34685 extid 35562 eusnsn 43255 |
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