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Mirrors > Home > MPE Home > Th. List > intexab | Structured version Visualization version GIF version |
Description: The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.) |
Ref | Expression |
---|---|
intexab | ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abn0 3908 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) | |
2 | intex 4747 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | |
3 | 1, 2 | bitr3i 265 | 1 ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∃wex 1695 ∈ wcel 1977 {cab 2596 ≠ wne 2780 Vcvv 3173 ∅c0 3874 ∩ 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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
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-ne 2782 df-ral 2901 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-int 4411 |
This theorem is referenced by: intexrab 4750 tcmin 8500 cfval 8952 efgval 17953 relintabex 36906 rclexi 36941 rtrclex 36943 trclexi 36946 rtrclexi 36947 |
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