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 4060 | . . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
2 | 1 | ex 450 | . . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)) |
3 | vex 3203 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | 3 | elint2 4482 | . . 3 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
5 | eluni2 4440 | . . 3 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
6 | 2, 4, 5 | 3imtr4g 285 | . 2 ⊢ (𝐴 ≠ ∅ → (𝑥 ∈ ∩ 𝐴 → 𝑥 ∈ ∪ 𝐴)) |
7 | 6 | ssrdv 3609 | 1 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 ∅c0 3915 ∪ cuni 4436 ∩ cint 4475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-uni 4437 df-int 4476 |
This theorem is referenced by: unissint 4501 intssuni2 4502 fin23lem31 9165 wunint 9537 tskint 9607 incexc 14569 incexc2 14570 subgint 17618 efgval 18130 lbsextlem3 19160 cssmre 20037 uffixfr 21727 uffix2 21728 uffixsn 21729 insiga 30200 dfon2lem8 31695 bj-intss 33053 intidl 33828 elrfi 37257 |
Copyright terms: Public domain | W3C validator |