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Mirrors > Home > MPE Home > Th. List > elintab | Structured version Visualization version GIF version |
Description: Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.) |
Ref | Expression |
---|---|
inteqab.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elintab | ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inteqab.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | elint 4416 | . 2 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦)) |
3 | nfsab1 2600 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | |
4 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦 | |
5 | 3, 4 | nfim 1813 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) |
6 | nfv 1830 | . . 3 ⊢ Ⅎ𝑦(𝜑 → 𝐴 ∈ 𝑥) | |
7 | eleq1 2676 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})) | |
8 | abid 2598 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
9 | 7, 8 | syl6bb 275 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)) |
10 | eleq2 2677 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥)) | |
11 | 9, 10 | imbi12d 333 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) ↔ (𝜑 → 𝐴 ∈ 𝑥))) |
12 | 5, 6, 11 | cbval 2259 | . 2 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)) |
13 | 2, 12 | bitri 263 | 1 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∈ wcel 1977 {cab 2596 Vcvv 3173 ∩ 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 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-v 3175 df-int 4411 |
This theorem is referenced by: elintrab 4423 intmin4 4441 intab 4442 intid 4853 dfom3 8427 dfom5 8430 tc2 8501 dfnn2 10910 brintclab 13590 efgi 17955 efgi2 17961 mclsax 30720 heibor1lem 32778 elmapintab 36921 intabssd 36935 cotrintab 36940 dffrege76 37253 |
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