Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > intexrab | Structured version Visualization version GIF version |
Description: The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.) |
Ref | Expression |
---|---|
intexrab | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intexab 4749 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V) | |
2 | df-rex 2902 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | df-rab 2905 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
4 | 3 | inteqi 4414 | . . 3 ⊢ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} = ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
5 | 4 | eleq1i 2679 | . 2 ⊢ (∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V) |
6 | 1, 2, 5 | 3bitr4i 291 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 {cab 2596 ∃wrex 2897 {crab 2900 Vcvv 3173 ∩ cint 4410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-int 4411 |
This theorem is referenced by: onintrab2 6894 rankf 8540 rankvalb 8543 cardf2 8652 tskmval 9540 lspval 18796 aspval 19149 clsval 20651 spanval 27576 rgspnval 36757 |
Copyright terms: Public domain | W3C validator |