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Mirrors > Home > MPE Home > Th. List > Mathboxes > elinintab | Structured version Visualization version GIF version |
Description: Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 13-Aug-2020.) |
Ref | Expression |
---|---|
elinintab | ⊢ (𝐴 ∈ (𝐵 ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3908 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ ∩ {𝑥 ∣ 𝜑})) | |
2 | elintabg 41220 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))) | |
3 | 2 | pm5.32i 576 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))) |
4 | 1, 3 | bitri 275 | 1 ⊢ (𝐴 ∈ (𝐵 ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1537 ∈ wcel 2104 {cab 2713 ∩ cin 3891 ∩ cint 4886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-tru 1542 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-v 3439 df-in 3899 df-int 4887 |
This theorem is referenced by: inintabss 41224 inintabd 41225 elcnvcnvintab 41228 |
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