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| Mirrors > Home > MPE Home > Th. List > intssuni | Structured version Visualization version GIF version | ||
| Description: The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.) |
| Ref | Expression |
|---|---|
| intssuni | ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r19.2z 4495 | . . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
| 2 | 1 | ex 412 | . . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)) |
| 3 | vex 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 4 | 3 | elint2 4953 | . . 3 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
| 5 | eluni2 4911 | . . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
| 6 | 2, 4, 5 | 3imtr4g 296 | . 2 ⊢ (𝐴 ≠ ∅ → (𝑥 ∈ ∩ 𝐴 → 𝑥 ∈ ∪ 𝐴)) |
| 7 | 6 | ssrdv 3989 | 1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ∅c0 4333 ∪ cuni 4907 ∩ cint 4946 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-ss 3968 df-nul 4334 df-uni 4908 df-int 4947 |
| This theorem is referenced by: unissint 4972 intssuni2 4973 intss2 5108 fin23lem31 10383 wunint 10755 tskint 10825 incexc 15873 incexc2 15874 subgint 19168 efgval 19735 lbsextlem3 21162 cssmre 21711 uffixfr 23931 uffix2 23932 uffixsn 23933 ssdifidllem 33484 ssmxidllem 33501 insiga 34138 dfon2lem8 35791 intidl 38036 elrfi 42705 toplatglb 48890 |
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