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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 5094 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (〈𝐴, 𝐵〉 = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))) | |
4 | 1, 2, 3 | mp2an 664 | 1 ⊢ (〈𝐴, 𝐵〉 = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 = wceq 1630 ∈ wcel 2144 Vcvv 3349 {csn 4314 〈cop 4320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-v 3351 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-sn 4315 df-pr 4317 df-op 4321 |
This theorem is referenced by: snopeqop 5098 propeqop 5100 relop 5411 |
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