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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 6454. (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 4592 | . . 3 ⊢ {𝑌} = {𝑥 ∣ 𝑥 = 𝑌} | |
2 | 1 | eqeq2i 2750 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑌}) |
3 | abbi 2809 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑌) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑌}) | |
4 | 2, 3 | bitr4i 278 | 1 ⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1540 = wceq 1542 {cab 2714 {csn 4591 |
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 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-sn 4592 |
This theorem is referenced by: rabeqsn 4632 euabsn2 4691 dfiota2 6454 dfaiota2 45392 aiotaval 45401 |
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