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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 5395 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (〈𝐴, 𝐵〉 = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (〈𝐴, 𝐵〉 = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 〈cop 4575 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 |
This theorem is referenced by: snopeqop 5398 propeqop 5399 relop 5723 |
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