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Mirrors > Home > MPE Home > Th. List > elintrab | Structured version Visualization version GIF version |
Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.) |
Ref | Expression |
---|---|
elintab.ex | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elintrab | ⊢ (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elintab.ex | . . . 4 ⊢ 𝐴 ∈ V | |
2 | 1 | elintab 4963 | . . 3 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝐴 ∈ 𝑥)) |
3 | impexp 450 | . . . 4 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝐴 ∈ 𝑥) ↔ (𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ∈ 𝑥))) | |
4 | 3 | albii 1816 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝐴 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ∈ 𝑥))) |
5 | 2, 4 | bitri 275 | . 2 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ∈ 𝑥))) |
6 | df-rab 3434 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
7 | 6 | inteqi 4955 | . . 3 ⊢ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} = ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} |
8 | 7 | eleq2i 2831 | . 2 ⊢ (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ ∩ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}) |
9 | df-ral 3060 | . 2 ⊢ (∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝜑 → 𝐴 ∈ 𝑥))) | |
10 | 5, 8, 9 | 3bitr4i 303 | 1 ⊢ (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 ∈ wcel 2106 {cab 2712 ∀wral 3059 {crab 3433 Vcvv 3478 ∩ 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-rex 3069 df-rab 3434 df-int 4952 |
This theorem is referenced by: elintrabg 4966 intmin 4973 naddelim 8723 rankunb 9888 isf34lem4 10415 ist1-3 23373 filufint 23944 elspani 31572 fldgensdrg 33296 evls1fldgencl 33695 ldsysgenld 34141 ldgenpisyslem1 34144 kur14lem9 35199 pclclN 39874 elpclN 39875 naddwordnexlem4 43391 lcosslsp 48284 |
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