![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 4666 | . . 3 ⊢ 𝐵 ∈ {𝐵} |
3 | eleq2 2814 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → (𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝐵 ∈ {𝐵})) | |
4 | 2, 3 | mpbiri 257 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
5 | elrabi 3673 | . 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 2098 {crab 3418 Vcvv 3461 {csn 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-sn 4631 |
This theorem is referenced by: ddemeas 33983 |
Copyright terms: Public domain | W3C validator |