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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 4882 | . 2 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦)) |
3 | nfsab1 2808 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | |
4 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑦 | |
5 | 3, 4 | nfim 1897 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) |
6 | nfv 1915 | . . 3 ⊢ Ⅎ𝑦(𝜑 → 𝐴 ∈ 𝑥) | |
7 | eleq1w 2895 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})) | |
8 | abid 2803 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
9 | 7, 8 | syl6bb 289 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)) |
10 | eleq2w 2896 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥)) | |
11 | 9, 10 | imbi12d 347 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) ↔ (𝜑 → 𝐴 ∈ 𝑥))) |
12 | 5, 6, 11 | cbvalv1 2361 | . 2 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ 𝑦) ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)) |
13 | 2, 12 | bitri 277 | 1 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∈ wcel 2114 {cab 2799 Vcvv 3494 ∩ cint 4876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-int 4877 |
This theorem is referenced by: elintrab 4888 intmin4 4905 intab 4906 intid 5350 dfom3 9110 dfom5 9113 tc2 9184 dfnn2 11651 brintclab 14361 efgi 18845 efgi2 18851 mclsax 32816 heibor1lem 35102 intabssd 39905 elmapintab 39976 cotrintab 39994 dffrege76 40305 |
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