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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 2823 | . 2 ⊢ 𝐴 = 𝐴 | |
2 | eqid 2823 | . 2 ⊢ {𝐴} = {𝐴} | |
3 | snopeqopsnid.a | . . 3 ⊢ 𝐴 ∈ V | |
4 | 3, 3 | snopeqop 5398 | . 2 ⊢ ({〈𝐴, 𝐴〉} = 〈{𝐴}, {𝐴}〉 ↔ (𝐴 = 𝐴 ∧ {𝐴} = {𝐴} ∧ {𝐴} = {𝐴})) |
5 | 1, 2, 2, 4 | mpbir3an 1337 | 1 ⊢ {〈𝐴, 𝐴〉} = 〈{𝐴}, {𝐴}〉 |
Colors of variables: wff setvar class |
Syntax hints: = 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: funsneqopb 6916 vtxvalsnop 26828 iedgvalsnop 26829 |
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