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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 4153 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) | |
2 | intex 5010 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | |
3 | 1, 2 | bitr3i 269 | 1 ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∃wex 1875 ∈ wcel 2157 {cab 2783 ≠ wne 2969 Vcvv 3383 ∅c0 4113 ∩ cint 4665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-v 3385 df-dif 3770 df-in 3774 df-ss 3781 df-nul 4114 df-int 4666 |
This theorem is referenced by: intexrab 5013 tcmin 8865 cfval 9355 efgval 18440 relintabex 38658 rclexi 38693 rtrclex 38695 trclexi 38698 rtrclexi 38699 aiotaexb 41926 |
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