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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfint3 | Structured version Visualization version GIF version |
Description: Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.) |
Ref | Expression |
---|---|
dfint3 | ⊢ ∩ 𝐴 = (V ∖ (◡(V ∖ E ) “ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfint2 4611 | . 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} | |
2 | ralnex 3140 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑦◡(V ∖ E )𝑥 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥) | |
3 | vex 3352 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
4 | vex 3352 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | brcnv 5443 | . . . . . . . 8 ⊢ (𝑦◡(V ∖ E )𝑥 ↔ 𝑥(V ∖ E )𝑦) |
6 | brv 5068 | . . . . . . . . 9 ⊢ 𝑥V𝑦 | |
7 | brdif 4837 | . . . . . . . . 9 ⊢ (𝑥(V ∖ E )𝑦 ↔ (𝑥V𝑦 ∧ ¬ 𝑥 E 𝑦)) | |
8 | 6, 7 | mpbiran 680 | . . . . . . . 8 ⊢ (𝑥(V ∖ E )𝑦 ↔ ¬ 𝑥 E 𝑦) |
9 | 5, 8 | bitr2i 265 | . . . . . . 7 ⊢ (¬ 𝑥 E 𝑦 ↔ 𝑦◡(V ∖ E )𝑥) |
10 | 9 | con1bii 345 | . . . . . 6 ⊢ (¬ 𝑦◡(V ∖ E )𝑥 ↔ 𝑥 E 𝑦) |
11 | epel 5165 | . . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
12 | 10, 11 | bitr2i 265 | . . . . 5 ⊢ (𝑥 ∈ 𝑦 ↔ ¬ 𝑦◡(V ∖ E )𝑥) |
13 | 12 | ralbii 3128 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡(V ∖ E )𝑥) |
14 | eldif 3731 | . . . . . 6 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (◡(V ∖ E ) “ 𝐴))) | |
15 | 4, 14 | mpbiran 680 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ¬ 𝑥 ∈ (◡(V ∖ E ) “ 𝐴)) |
16 | 4 | elima 5612 | . . . . 5 ⊢ (𝑥 ∈ (◡(V ∖ E ) “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥) |
17 | 15, 16 | xchbinx 323 | . . . 4 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ¬ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥) |
18 | 2, 13, 17 | 3bitr4ri 293 | . . 3 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
19 | 18 | abbi2i 2886 | . 2 ⊢ (V ∖ (◡(V ∖ E ) “ 𝐴)) = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} |
20 | 1, 19 | eqtr4i 2795 | 1 ⊢ ∩ 𝐴 = (V ∖ (◡(V ∖ E ) “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1630 ∈ wcel 2144 {cab 2756 ∀wral 3060 ∃wrex 3061 Vcvv 3349 ∖ cdif 3718 ∩ cint 4609 class class class wbr 4784 E cep 5161 ◡ccnv 5248 “ cima 5252 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-ral 3065 df-rex 3066 df-rab 3069 df-v 3351 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-sn 4315 df-pr 4317 df-op 4321 df-int 4610 df-br 4785 df-opab 4845 df-eprel 5162 df-xp 5255 df-cnv 5257 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 |
This theorem is referenced by: (None) |
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