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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 4290 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑) | |
2 | intex 5204 | . 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | |
3 | 1, 2 | bitr3i 280 | 1 ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∃wex 1781 ∈ wcel 2111 {cab 2776 ≠ wne 2987 Vcvv 3441 ∅c0 4243 ∩ cint 4838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rab 3115 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-int 4839 |
This theorem is referenced by: intexrab 5207 tcmin 9167 cfval 9658 efgval 18835 relintabex 40281 rclexi 40315 rtrclex 40317 trclexi 40320 rtrclexi 40321 aiotaexb 43646 |
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