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Mirrors > Home > ILE Home > Th. List > intexrabim | GIF version |
Description: The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.) |
Ref | Expression |
---|---|
intexrabim | ⊢ (∃x ∈ A φ → ∩ {x ∈ A ∣ φ} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intexabim 3897 | . 2 ⊢ (∃x(x ∈ A ∧ φ) → ∩ {x ∣ (x ∈ A ∧ φ)} ∈ V) | |
2 | df-rex 2306 | . 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ)) | |
3 | df-rab 2309 | . . . 4 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)} | |
4 | 3 | inteqi 3610 | . . 3 ⊢ ∩ {x ∈ A ∣ φ} = ∩ {x ∣ (x ∈ A ∧ φ)} |
5 | 4 | eleq1i 2100 | . 2 ⊢ (∩ {x ∈ A ∣ φ} ∈ V ↔ ∩ {x ∣ (x ∈ A ∧ φ)} ∈ V) |
6 | 1, 2, 5 | 3imtr4i 190 | 1 ⊢ (∃x ∈ A φ → ∩ {x ∈ A ∣ φ} ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∃wex 1378 ∈ wcel 1390 {cab 2023 ∃wrex 2301 {crab 2304 Vcvv 2551 ∩ cint 3606 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-in 2918 df-ss 2925 df-int 3607 |
This theorem is referenced by: (None) |
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