![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > absn | Structured version Visualization version GIF version |
Description: Condition for a class abstraction to be a singleton. Formerly part of proof of dfiota2 6284. (Contributed by Andrew Salmon, 30-Jun-2011.) (Revised by AV, 24-Aug-2022.) |
Ref | Expression |
---|---|
absn | ⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sn 4526 | . . 3 ⊢ {𝑌} = {𝑥 ∣ 𝑥 = 𝑌} | |
2 | 1 | eqeq2i 2811 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑌}) |
3 | abbi 2865 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑌) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑌}) | |
4 | 2, 3 | bitr4i 281 | 1 ⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wal 1536 = wceq 1538 {cab 2776 {csn 4525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-sn 4526 |
This theorem is referenced by: rabeqsn 4566 euabsn2 4621 dfiota2 6284 dfaiota2 43643 aiotaval 43650 |
Copyright terms: Public domain | W3C validator |