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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 4880 | . 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} | |
2 | ralnex 3238 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑦◡(V ∖ E )𝑥 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥) | |
3 | vex 3499 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
4 | vex 3499 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | brcnv 5755 | . . . . . . . 8 ⊢ (𝑦◡(V ∖ E )𝑥 ↔ 𝑥(V ∖ E )𝑦) |
6 | brv 5366 | . . . . . . . . 9 ⊢ 𝑥V𝑦 | |
7 | brdif 5121 | . . . . . . . . 9 ⊢ (𝑥(V ∖ E )𝑦 ↔ (𝑥V𝑦 ∧ ¬ 𝑥 E 𝑦)) | |
8 | 6, 7 | mpbiran 707 | . . . . . . . 8 ⊢ (𝑥(V ∖ E )𝑦 ↔ ¬ 𝑥 E 𝑦) |
9 | 5, 8 | bitr2i 278 | . . . . . . 7 ⊢ (¬ 𝑥 E 𝑦 ↔ 𝑦◡(V ∖ E )𝑥) |
10 | 9 | con1bii 359 | . . . . . 6 ⊢ (¬ 𝑦◡(V ∖ E )𝑥 ↔ 𝑥 E 𝑦) |
11 | epel 5471 | . . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
12 | 10, 11 | bitr2i 278 | . . . . 5 ⊢ (𝑥 ∈ 𝑦 ↔ ¬ 𝑦◡(V ∖ E )𝑥) |
13 | 12 | ralbii 3167 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡(V ∖ E )𝑥) |
14 | eldif 3948 | . . . . . 6 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (◡(V ∖ E ) “ 𝐴))) | |
15 | 4, 14 | mpbiran 707 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ¬ 𝑥 ∈ (◡(V ∖ E ) “ 𝐴)) |
16 | 4 | elima 5936 | . . . . 5 ⊢ (𝑥 ∈ (◡(V ∖ E ) “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥) |
17 | 15, 16 | xchbinx 336 | . . . 4 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ¬ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥) |
18 | 2, 13, 17 | 3bitr4ri 306 | . . 3 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
19 | 18 | abbi2i 2955 | . 2 ⊢ (V ∖ (◡(V ∖ E ) “ 𝐴)) = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} |
20 | 1, 19 | eqtr4i 2849 | 1 ⊢ ∩ 𝐴 = (V ∖ (◡(V ∖ E ) “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 {cab 2801 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ∖ cdif 3935 ∩ cint 4878 class class class wbr 5068 E cep 5466 ◡ccnv 5556 “ cima 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-int 4879 df-br 5069 df-opab 5131 df-eprel 5467 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 |
This theorem is referenced by: (None) |
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