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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 2735 | . 2 ⊢ 𝐴 = 𝐴 | |
2 | eqid 2735 | . 2 ⊢ {𝐴} = {𝐴} | |
3 | snopeqopsnid.a | . . 3 ⊢ 𝐴 ∈ V | |
4 | 3, 3 | snopeqop 5516 | . 2 ⊢ ({〈𝐴, 𝐴〉} = 〈{𝐴}, {𝐴}〉 ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴})) |
5 | 1, 2, 2, 4 | mpbir3an 1340 | 1 ⊢ {〈𝐴, 𝐴〉} = 〈{𝐴}, {𝐴}〉 |
Colors of variables: wff setvar class |
Syntax hints: = 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: funsneqopb 7172 vtxvalsnop 29073 iedgvalsnop 29074 |
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