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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 4391 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) | |
2 | intex 5350 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | |
3 | 1, 2 | bitr3i 277 | 1 ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∃wex 1776 ∈ wcel 2106 {cab 2712 ≠ wne 2938 Vcvv 3478 ∅c0 4339 ∩ 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 ax-sep 5302 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-in 3970 df-ss 3980 df-nul 4340 df-int 4952 |
This theorem is referenced by: intexrab 5353 tcmin 9779 cfval 10285 efgval 19750 relintabex 43571 rclexi 43605 rtrclex 43607 trclexi 43610 rtrclexi 43611 aiotaexb 47039 |
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