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Mirrors > Home > MPE Home > Th. List > snopeqopsnid | Structured version Visualization version GIF version |
Description: Equivalence for an ordered pair of two identical singletons equal to a singleton of an ordered pair. (Contributed by AV, 24-Sep-2020.) (Revised by AV, 15-Jul-2022.) (Avoid depending on this detail.) |
Ref | Expression |
---|---|
snopeqopsnid.a | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
snopeqopsnid | ⊢ {〈𝐴, 𝐴〉} = 〈{𝐴}, {𝐴}〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . 2 ⊢ 𝐴 = 𝐴 | |
2 | eqid 2736 | . 2 ⊢ {𝐴} = {𝐴} | |
3 | snopeqopsnid.a | . . 3 ⊢ 𝐴 ∈ V | |
4 | 3, 3 | snopeqop 5462 | . 2 ⊢ ({〈𝐴, 𝐴〉} = 〈{𝐴}, {𝐴}〉 ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴})) |
5 | 1, 2, 2, 4 | mpbir3an 1341 | 1 ⊢ {〈𝐴, 𝐴〉} = 〈{𝐴}, {𝐴}〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3444 {csn 4585 〈cop 4591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2943 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 |
This theorem is referenced by: funsneqopb 7095 vtxvalsnop 27890 iedgvalsnop 27891 |
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