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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 4012 | . . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
2 | 1 | ex 449 | . . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)) |
3 | vex 3176 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | 3 | elint2 4417 | . . 3 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
5 | eluni2 4376 | . . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
6 | 2, 4, 5 | 3imtr4g 284 | . 2 ⊢ (𝐴 ≠ ∅ → (𝑥 ∈ ∩ 𝐴 → 𝑥 ∈ ∪ 𝐴)) |
7 | 6 | ssrdv 3574 | 1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 ∅c0 3874 ∪ cuni 4372 ∩ cint 4410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-uni 4373 df-int 4411 |
This theorem is referenced by: unissint 4436 intssuni2 4437 fin23lem31 9048 wunint 9416 tskint 9486 incexc 14408 incexc2 14409 subgint 17441 efgval 17953 lbsextlem3 18981 cssmre 19856 uffixfr 21537 uffix2 21538 uffixsn 21539 insiga 29527 dfon2lem8 30939 intidl 32998 elrfi 36275 |
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