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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 5205 | . . . 4 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∣ 𝜑} = V) | |
2 | nrelv 5637 | . . . . 5 ⊢ ¬ Rel V | |
3 | releq 5615 | . . . . 5 ⊢ (∩ {𝑥 ∣ 𝜑} = V → (Rel ∩ {𝑥 ∣ 𝜑} ↔ Rel V)) | |
4 | 2, 3 | mtbiri 330 | . . . 4 ⊢ (∩ {𝑥 ∣ 𝜑} = V → ¬ Rel ∩ {𝑥 ∣ 𝜑}) |
5 | 1, 4 | sylbi 220 | . . 3 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V → ¬ Rel ∩ {𝑥 ∣ 𝜑}) |
6 | 5 | con4i 114 | . 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ∈ V) |
7 | intexab 5206 | . 2 ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | |
8 | 6, 7 | sylibr 237 | 1 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2776 Vcvv 3441 ∩ cint 4838 Rel wrel 5524 |
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 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 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-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-int 4839 df-opab 5093 df-xp 5525 df-rel 5526 |
This theorem is referenced by: relintab 40283 |
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