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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 RP, 13-Aug-2020.) |
Ref | Expression |
---|---|
elintabg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elintg 4846 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ∈ 𝑦)) | |
2 | eleq2w 2873 | . . 3 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥)) | |
3 | 2 | ralab2 3636 | . 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ∈ 𝑦 ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)) |
4 | 1, 3 | syl6bb 290 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 ∈ wcel 2111 {cab 2776 ∀wral 3106 ∩ cint 4838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-int 4839 |
This theorem is referenced by: elinintab 40275 |
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