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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 4335 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) | |
2 | intex 5232 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | |
3 | 1, 2 | bitr3i 279 | 1 ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∃wex 1776 ∈ wcel 2110 {cab 2799 ≠ wne 3016 Vcvv 3494 ∅c0 4290 ∩ cint 4868 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-in 3942 df-ss 3951 df-nul 4291 df-int 4869 |
This theorem is referenced by: intexrab 5235 tcmin 9177 cfval 9663 efgval 18837 relintabex 39934 rclexi 39968 rtrclex 39970 trclexi 39973 rtrclexi 39974 aiotaexb 43283 |
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