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Mirrors > Home > MPE Home > Th. List > Mathboxes > rabsnel | Structured version Visualization version GIF version |
Description: Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by Thierry Arnoux, 15-Sep-2018.) |
Ref | Expression |
---|---|
rabsnel.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
rabsnel | ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabsnel.1 | . . . 4 ⊢ 𝐵 ∈ V | |
2 | 1 | snid 4594 | . . 3 ⊢ 𝐵 ∈ {𝐵} |
3 | eleq2 2901 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝐵 ∈ {𝐵})) | |
4 | 2, 3 | mpbiri 260 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
5 | elrabi 3674 | . 2 ⊢ (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝐵 ∈ 𝐴) | |
6 | 4, 5 | syl 17 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 {csn 4560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-sn 4561 |
This theorem is referenced by: ddemeas 31490 |
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