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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 4284 | . . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
2 | 1 | ex 403 | . . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)) |
3 | vex 3417 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | 3 | elint2 4706 | . . 3 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
5 | eluni2 4664 | . . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
6 | 2, 4, 5 | 3imtr4g 288 | . 2 ⊢ (𝐴 ≠ ∅ → (𝑥 ∈ ∩ 𝐴 → 𝑥 ∈ ∪ 𝐴)) |
7 | 6 | ssrdv 3833 | 1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2164 ≠ wne 2999 ∀wral 3117 ∃wrex 3118 ⊆ wss 3798 ∅c0 4146 ∪ cuni 4660 ∩ cint 4699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-v 3416 df-dif 3801 df-in 3805 df-ss 3812 df-nul 4147 df-uni 4661 df-int 4700 |
This theorem is referenced by: unissint 4723 intssuni2 4724 fin23lem31 9487 wunint 9859 tskint 9929 incexc 14950 incexc2 14951 subgint 17976 efgval 18488 lbsextlem3 19528 cssmre 20407 uffixfr 22104 uffix2 22105 uffixsn 22106 insiga 30741 dfon2lem8 32228 bj-intss 33575 intidl 34369 elrfi 38100 |
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