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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 3709 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥) | |
2 | vex 2623 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | ssex 3982 | . . 3 ⊢ (∩ 𝐴 ⊆ 𝑥 → ∩ 𝐴 ∈ V) |
4 | 1, 3 | syl 14 | . 2 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V) |
5 | 4 | exlimiv 1535 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃wex 1427 ∈ wcel 1439 Vcvv 2620 ⊆ wss 3000 ∩ cint 3694 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-in 3006 df-ss 3013 df-int 3695 |
This theorem is referenced by: intexabim 3994 iinexgm 3996 onintonm 4347 |
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