Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > clabel | Structured version Visualization version GIF version |
Description: Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.) |
Ref | Expression |
---|---|
clabel | ⊢ ({𝑥 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfclel 2810 | . 2 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝐴)) | |
2 | abeq2 2862 | . . . 4 ⊢ (𝑦 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)) | |
3 | 2 | anbi2ci 628 | . . 3 ⊢ ((𝑦 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))) |
4 | 3 | exbii 1855 | . 2 ⊢ (∃𝑦(𝑦 = {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))) |
5 | 1, 4 | bitri 278 | 1 ⊢ ({𝑥 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∀wal 1541 = wceq 1543 ∃wex 1787 ∈ wcel 2112 {cab 2714 |
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-8 2114 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-clel 2809 |
This theorem is referenced by: sbabel 2931 grothprimlem 10412 ntrneiel2 41314 |
Copyright terms: Public domain | W3C validator |