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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 5429 | . 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 1539 ∈ wcel 2104 Vcvv 3437 {csn 4565 ⟨cop 4571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pr 5361 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 |
This theorem is referenced by: funsneqopb 7052 vtxvalsnop 27452 iedgvalsnop 27453 |
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