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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 5045 | . . . 4 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∣ 𝜑} = V) | |
2 | 0nelxp 5380 | . . . . . . 7 ⊢ ¬ ∅ ∈ (V × V) | |
3 | 0ex 5016 | . . . . . . . 8 ⊢ ∅ ∈ V | |
4 | eleq1 2894 | . . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V))) | |
5 | 4 | notbid 310 | . . . . . . . 8 ⊢ (𝑥 = ∅ → (¬ 𝑥 ∈ (V × V) ↔ ¬ ∅ ∈ (V × V))) |
6 | 3, 5 | spcev 3517 | . . . . . . 7 ⊢ (¬ ∅ ∈ (V × V) → ∃𝑥 ¬ 𝑥 ∈ (V × V)) |
7 | 2, 6 | ax-mp 5 | . . . . . 6 ⊢ ∃𝑥 ¬ 𝑥 ∈ (V × V) |
8 | nss 3888 | . . . . . . . 8 ⊢ (¬ V ⊆ (V × V) ↔ ∃𝑥(𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V × V))) | |
9 | df-rex 3123 | . . . . . . . 8 ⊢ (∃𝑥 ∈ V ¬ 𝑥 ∈ (V × V) ↔ ∃𝑥(𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V × V))) | |
10 | rexv 3437 | . . . . . . . 8 ⊢ (∃𝑥 ∈ V ¬ 𝑥 ∈ (V × V) ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V)) | |
11 | 8, 9, 10 | 3bitr2i 291 | . . . . . . 7 ⊢ (¬ V ⊆ (V × V) ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V)) |
12 | df-rel 5353 | . . . . . . 7 ⊢ (Rel V ↔ V ⊆ (V × V)) | |
13 | 11, 12 | xchnxbir 325 | . . . . . 6 ⊢ (¬ Rel V ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V)) |
14 | 7, 13 | mpbir 223 | . . . . 5 ⊢ ¬ Rel V |
15 | releq 5440 | . . . . 5 ⊢ (∩ {𝑥 ∣ 𝜑} = V → (Rel ∩ {𝑥 ∣ 𝜑} ↔ Rel V)) | |
16 | 14, 15 | mtbiri 319 | . . . 4 ⊢ (∩ {𝑥 ∣ 𝜑} = V → ¬ Rel ∩ {𝑥 ∣ 𝜑}) |
17 | 1, 16 | sylbi 209 | . . 3 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V → ¬ Rel ∩ {𝑥 ∣ 𝜑}) |
18 | 17 | con4i 114 | . 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ∈ V) |
19 | intexab 5046 | . 2 ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V) | |
20 | 18, 19 | sylibr 226 | 1 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1656 ∃wex 1878 ∈ wcel 2164 {cab 2811 ∃wrex 3118 Vcvv 3414 ⊆ wss 3798 ∅c0 4146 ∩ cint 4699 × cxp 5344 Rel wrel 5351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-int 4700 df-opab 4938 df-xp 5352 df-rel 5353 |
This theorem is referenced by: relintab 38729 |
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