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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 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 904 | . . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐴) ↔ 𝑥 = 𝐴) | |
| 2 | 1 | abbii 2808 | . 2 ⊢ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)} = {𝑥 ∣ 𝑥 = 𝐴} | 
| 3 | dfpr2 4645 | . 2 ⊢ {𝐴, 𝐴} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)} | |
| 4 | df-sn 4626 | . 2 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴} | |
| 5 | 2, 3, 4 | 3eqtr4ri 2775 | 1 ⊢ {𝐴} = {𝐴, 𝐴} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∨ wo 847 = wceq 1539 {cab 2713 {csn 4625 {cpr 4627 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-un 3955 df-sn 4626 df-pr 4628 | 
| This theorem is referenced by: (None) | 
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