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Mirrors > Home > MPE Home > Th. List > elintabg | Structured version Visualization version GIF version |
Description: Two ways of saying a set is an element of the intersection of a class. (Contributed by NM, 30-Aug-1993.) Put in closed form. (Revised by RP, 13-Aug-2020.) |
Ref | Expression |
---|---|
elintabg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elintg 4959 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ∈ 𝑦)) | |
2 | eleq2w 2823 | . . 3 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥)) | |
3 | 2 | ralab2 3706 | . 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ∈ 𝑦 ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)) |
4 | 1, 3 | bitrdi 287 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∩ cint 4951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-int 4952 |
This theorem is referenced by: elintab 4963 intidg 5468 elinintab 43565 |
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