![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5513 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (〈𝐴, 𝐵〉 = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (〈𝐴, 𝐵〉 = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 〈cop 4637 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 |
This theorem is referenced by: snopeqop 5516 propeqop 5517 relop 5864 |
Copyright terms: Public domain | W3C validator |