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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 903 | . . 3 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐴) ↔ 𝑥 = 𝐴) | |
2 | 1 | abbii 2798 | . 2 ⊢ {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)} = {𝑥 ∣ 𝑥 = 𝐴} |
3 | dfpr2 4644 | . 2 ⊢ {𝐴, 𝐴} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐴)} | |
4 | df-sn 4626 | . 2 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴} | |
5 | 2, 3, 4 | 3eqtr4ri 2767 | 1 ⊢ {𝐴} = {𝐴, 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 846 = wceq 1534 {cab 2705 {csn 4625 {cpr 4627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3472 df-un 3950 df-sn 4626 df-pr 4628 |
This theorem is referenced by: (None) |
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