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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 5810 | . . . . 5 ⊢ ¬ Rel V | |
| 3 | releq 5786 | . . . . 5 ⊢ (∩ {𝑥 ∣ 𝜑} = V → (Rel ∩ {𝑥 ∣ 𝜑} ↔ Rel V)) | |
| 4 | 2, 3 | mtbiri 327 | . . . 4 ⊢ (∩ {𝑥 ∣ 𝜑} = V → ¬ Rel ∩ {𝑥 ∣ 𝜑}) | 
| 5 | 1, 4 | sylbi 217 | . . 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 234 | 1 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 Vcvv 3480 ∩ cint 4946 Rel wrel 5690 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-int 4947 df-opab 5206 df-xp 5691 df-rel 5692 | 
| This theorem is referenced by: relintab 43596 | 
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