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Mirrors > Home > MPE Home > Th. List > dfsn2ALT | Structured version Visualization version GIF version |
Description: Alternate definition of singleton, based on the (alternate) definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by AV, 12-Jun-2022.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
dfsn2ALT | ⊢ {𝐴} = {𝐴, 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oridm 901 | . . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐴) ↔ 𝑥 = 𝐴) | |
2 | 1 | abbii 2888 | . 2 ⊢ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)} = {𝑥 ∣ 𝑥 = 𝐴} |
3 | dfpr2 4588 | . 2 ⊢ {𝐴, 𝐴} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)} | |
4 | df-sn 4570 | . 2 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴} | |
5 | 2, 3, 4 | 3eqtr4ri 2857 | 1 ⊢ {𝐴} = {𝐴, 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 843 = wceq 1537 {cab 2801 {csn 4569 {cpr 4571 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-v 3498 df-un 3943 df-sn 4570 df-pr 4572 |
This theorem is referenced by: (None) |
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