![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dfsn2ALT | Structured version Visualization version GIF version |
Description: Alternate definition of singleton, based on the (alternate) definition of 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 2796 | . 2 ⊢ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)} = {𝑥 ∣ 𝑥 = 𝐴} |
3 | dfpr2 4642 | . 2 ⊢ {𝐴, 𝐴} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)} | |
4 | df-sn 4624 | . 2 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴} | |
5 | 2, 3, 4 | 3eqtr4ri 2765 | 1 ⊢ {𝐴} = {𝐴, 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 844 = wceq 1533 {cab 2703 {csn 4623 {cpr 4625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-un 3948 df-sn 4624 df-pr 4626 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |