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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 5351 | . . . 4 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∣ 𝜑} = V) | |
2 | nrelv 5813 | . . . . 5 ⊢ ¬ Rel V | |
3 | releq 5789 | . . . . 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 5352 | . 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 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 Vcvv 3478 ∩ cint 4951 Rel wrel 5694 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-int 4952 df-opab 5211 df-xp 5695 df-rel 5696 |
This theorem is referenced by: relintab 43573 |
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