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Mirrors > Home > MPE Home > Th. List > Mathboxes > relintabex | Structured version Visualization version GIF version |
Description: If the intersection of a class is a relation, then the class is nonempty. (Contributed by RP, 12-Aug-2020.) |
Ref | Expression |
---|---|
relintabex | ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intnex 5345 | . . . 4 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∣ 𝜑} = V) | |
2 | nrelv 5806 | . . . . 5 ⊢ ¬ Rel V | |
3 | releq 5782 | . . . . 5 ⊢ (∩ {𝑥 ∣ 𝜑} = V → (Rel ∩ {𝑥 ∣ 𝜑} ↔ Rel V)) | |
4 | 2, 3 | mtbiri 326 | . . . 4 ⊢ (∩ {𝑥 ∣ 𝜑} = V → ¬ Rel ∩ {𝑥 ∣ 𝜑}) |
5 | 1, 4 | sylbi 216 | . . 3 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V → ¬ Rel ∩ {𝑥 ∣ 𝜑}) |
6 | 5 | con4i 114 | . 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ∈ V) |
7 | intexab 5346 | . 2 ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | |
8 | 6, 7 | sylibr 233 | 1 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ∃wex 1774 ∈ wcel 2099 {cab 2703 Vcvv 3462 ∩ cint 4954 Rel wrel 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-int 4955 df-opab 5216 df-xp 5688 df-rel 5689 |
This theorem is referenced by: relintab 43250 |
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