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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iinssiun | Structured version Visualization version GIF version |
Description: An indexed intersection is a subset of the corresponding indexed union. (Contributed by Thierry Arnoux, 31-Dec-2021.) |
Ref | Expression |
---|---|
iinssiun | ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.2z 4284 | . . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
2 | 1 | ex 403 | . . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) |
3 | vex 3417 | . . . 4 ⊢ 𝑦 ∈ V | |
4 | eliin 4747 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) |
6 | eliun 4746 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
7 | 2, 5, 6 | 3imtr4g 288 | . 2 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)) |
8 | 7 | ssrdv 3833 | 1 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2164 ≠ wne 2999 ∀wral 3117 ∃wrex 3118 Vcvv 3414 ⊆ wss 3798 ∅c0 4146 ∪ ciun 4742 ∩ ciin 4743 |
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-iun 4744 df-iin 4745 |
This theorem is referenced by: (None) |
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