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Mirrors > Home > ILE Home > Th. List > inteximm | GIF version |
Description: The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.) |
Ref | Expression |
---|---|
inteximm | ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intss1 3630 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥) | |
2 | vex 2560 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | ssex 3894 | . . 3 ⊢ (∩ 𝐴 ⊆ 𝑥 → ∩ 𝐴 ∈ V) |
4 | 1, 3 | syl 14 | . 2 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V) |
5 | 4 | exlimiv 1489 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃wex 1381 ∈ wcel 1393 Vcvv 2557 ⊆ wss 2917 ∩ cint 3615 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-in 2924 df-ss 2931 df-int 3616 |
This theorem is referenced by: intexabim 3906 iinexgm 3908 onintonm 4243 |
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