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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 3794 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥) | |
2 | vex 2692 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | ssex 4073 | . . 3 ⊢ (∩ 𝐴 ⊆ 𝑥 → ∩ 𝐴 ∈ V) |
4 | 1, 3 | syl 14 | . 2 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V) |
5 | 4 | exlimiv 1578 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃wex 1469 ∈ wcel 1481 Vcvv 2689 ⊆ wss 3076 ∩ cint 3779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-in 3082 df-ss 3089 df-int 3780 |
This theorem is referenced by: intexabim 4085 iinexgm 4087 onintonm 4441 elfi2 6868 elfir 6869 fifo 6876 |
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