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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 3874 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥) | |
2 | vex 2755 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | ssex 4155 | . . 3 ⊢ (∩ 𝐴 ⊆ 𝑥 → ∩ 𝐴 ∈ V) |
4 | 1, 3 | syl 14 | . 2 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V) |
5 | 4 | exlimiv 1609 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃wex 1503 ∈ wcel 2160 Vcvv 2752 ⊆ wss 3144 ∩ cint 3859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-sep 4136 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-ss 3157 df-int 3860 |
This theorem is referenced by: intexabim 4170 iinexgm 4172 onintonm 4534 elfi2 7000 elfir 7001 fifo 7008 |
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