Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4442 | . . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
2 | 1 | ex 415 | . . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)) |
3 | vex 3499 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | 3 | elint2 4885 | . . 3 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
5 | eluni2 4844 | . . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
6 | 2, 4, 5 | 3imtr4g 298 | . 2 ⊢ (𝐴 ≠ ∅ → (𝑥 ∈ ∩ 𝐴 → 𝑥 ∈ ∪ 𝐴)) |
7 | 6 | ssrdv 3975 | 1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 ∅c0 4293 ∪ cuni 4840 ∩ cint 4878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-nul 4294 df-uni 4841 df-int 4879 |
This theorem is referenced by: unissint 4902 intssuni2 4903 fin23lem31 9767 wunint 10139 tskint 10209 incexc 15194 incexc2 15195 subgint 18305 efgval 18845 lbsextlem3 19934 cssmre 20839 uffixfr 22533 uffix2 22534 uffixsn 22535 ssmxidllem 30980 insiga 31398 dfon2lem8 33037 bj-intss 34393 intidl 35309 elrfi 39298 |
Copyright terms: Public domain | W3C validator |