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Mirrors > Home > MPE Home > Th. List > rabeqsn | Structured version Visualization version GIF version |
Description: Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 26-Aug-2022.) |
Ref | Expression |
---|---|
rabeqsn | ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑋} ↔ ∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ 𝑥 = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3060 | . . 3 ⊢ {𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} | |
2 | 1 | eqeq1i 2741 | . 2 ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑋} ↔ {𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} = {𝑋}) |
3 | absn 4545 | . 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑉 ∧ 𝜑)} = {𝑋} ↔ ∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ 𝑥 = 𝑋)) | |
4 | 2, 3 | bitri 278 | 1 ⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑋} ↔ ∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ 𝑥 = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∀wal 1541 = wceq 1543 ∈ wcel 2112 {cab 2714 {crab 3055 {csn 4527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-rab 3060 df-sn 4528 |
This theorem is referenced by: rabsn 4623 umgr2v2enb1 27568 clwwlknon1loop 28135 wlkl0 28404 rabeqsnd 30521 zarclssn 31491 made0 33743 k0004val0 41382 |
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