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Mirrors > Home > MPE Home > Th. List > Mathboxes > inintabss | Structured version Visualization version GIF version |
Description: Upper bound on intersection of class and the intersection of a class. (Contributed by RP, 13-Aug-2020.) |
Ref | Expression |
---|---|
inintabss | ⊢ (𝐴 ∩ ∩ {𝑥 ∣ 𝜑}) ⊆ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . . . 4 ⊢ (𝑢 ∈ 𝐴 → (∃𝑥𝜑 → 𝑢 ∈ 𝐴)) | |
2 | 1 | anim1i 590 | . . 3 ⊢ ((𝑢 ∈ 𝐴 ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥)) → ((∃𝑥𝜑 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥))) |
3 | elinintab 36900 | . . 3 ⊢ (𝑢 ∈ (𝐴 ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝑢 ∈ 𝐴 ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥))) | |
4 | vex 3176 | . . . 4 ⊢ 𝑢 ∈ V | |
5 | elinintrab 36902 | . . . 4 ⊢ (𝑢 ∈ V → (𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥)))) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ (𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥))) |
7 | 2, 3, 6 | 3imtr4i 280 | . 2 ⊢ (𝑢 ∈ (𝐴 ∩ ∩ {𝑥 ∣ 𝜑}) → 𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)}) |
8 | 7 | ssriv 3572 | 1 ⊢ (𝐴 ∩ ∩ {𝑥 ∣ 𝜑}) ⊆ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 {crab 2900 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ∩ 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 ax-sep 4709 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 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-ral 2901 df-rab 2905 df-v 3175 df-in 3547 df-ss 3554 df-pw 4110 df-int 4411 |
This theorem is referenced by: (None) |
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