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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 5447 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))) | |
4 | 1, 2, 3 | mp2an 689 | 1 ⊢ (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 Vcvv 3441 {csn 4573 ⟨cop 4579 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2941 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 |
This theorem is referenced by: snopeqop 5450 propeqop 5451 relop 5792 |
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