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Mirrors > Home > MPE Home > Th. List > opeqsn | Structured version Visualization version GIF version |
Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Revised by AV, 15-Jul-2022.) (Avoid depending on this detail.) |
Ref | Expression |
---|---|
opeqsn.1 | ⊢ 𝐴 ∈ V |
opeqsn.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opeqsn | ⊢ (〈𝐴, 𝐵〉 = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeqsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | opeqsn.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | opeqsng 5157 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (〈𝐴, 𝐵〉 = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))) | |
4 | 1, 2, 3 | mp2an 684 | 1 ⊢ (〈𝐴, 𝐵〉 = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 Vcvv 3385 {csn 4368 〈cop 4374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 |
This theorem is referenced by: snopeqop 5161 propeqop 5163 relop 5476 |
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