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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 3781 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥) | |
2 | vex 2684 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | ssex 4060 | . . 3 ⊢ (∩ 𝐴 ⊆ 𝑥 → ∩ 𝐴 ∈ V) |
4 | 1, 3 | syl 14 | . 2 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V) |
5 | 4 | exlimiv 1577 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃wex 1468 ∈ wcel 1480 Vcvv 2681 ⊆ wss 3066 ∩ cint 3766 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-ss 3079 df-int 3767 |
This theorem is referenced by: intexabim 4072 iinexgm 4074 onintonm 4428 elfi2 6853 elfir 6854 fifo 6861 |
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