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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 6391. (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 4568 | . . 3 ⊢ {𝑌} = {𝑥 ∣ 𝑥 = 𝑌} | |
2 | 1 | eqeq2i 2753 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑌}) |
3 | abbi 2812 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑌) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑌}) | |
4 | 2, 3 | bitr4i 277 | 1 ⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1540 = wceq 1542 {cab 2717 {csn 4567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2072 df-clab 2718 df-cleq 2732 df-sn 4568 |
This theorem is referenced by: rabeqsn 4608 euabsn2 4667 dfiota2 6391 dfaiota2 44546 aiotaval 44555 |
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