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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 |
---|---|
elintab.ex | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elintab | ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elintab.ex | . 2 ⊢ 𝐴 ∈ V | |
2 | elintabg 4962 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (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 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-int 4952 |
This theorem is referenced by: elintrab 4965 intmin4 4982 intab 4983 intidOLD 5469 dfom3 9685 dfom5 9688 tc2 9780 dfnn2 12277 brintclab 15037 efgi 19752 efgi2 19758 dfn0s2 28351 mclsax 35554 heibor1lem 37796 intabssd 43509 elmapintab 43586 cotrintab 43604 dffrege76 43929 |
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