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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 902 | . . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐴) ↔ 𝑥 = 𝐴) | |
2 | 1 | abbii 2808 | . 2 ⊢ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)} = {𝑥 ∣ 𝑥 = 𝐴} |
3 | dfpr2 4581 | . 2 ⊢ {𝐴, 𝐴} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)} | |
4 | df-sn 4563 | . 2 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴} | |
5 | 2, 3, 4 | 3eqtr4ri 2777 | 1 ⊢ {𝐴} = {𝐴, 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 844 = wceq 1539 {cab 2715 {csn 4562 {cpr 4564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3432 df-un 3892 df-sn 4563 df-pr 4565 |
This theorem is referenced by: (None) |
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