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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 4840 | . 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} | |
2 | ralnex 3199 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑦◡(V ∖ E )𝑥 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥) | |
3 | vex 3444 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
4 | vex 3444 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | brcnv 5717 | . . . . . . . 8 ⊢ (𝑦◡(V ∖ E )𝑥 ↔ 𝑥(V ∖ E )𝑦) |
6 | brv 5329 | . . . . . . . . 9 ⊢ 𝑥V𝑦 | |
7 | brdif 5083 | . . . . . . . . 9 ⊢ (𝑥(V ∖ E )𝑦 ↔ (𝑥V𝑦 ∧ ¬ 𝑥 E 𝑦)) | |
8 | 6, 7 | mpbiran 708 | . . . . . . . 8 ⊢ (𝑥(V ∖ E )𝑦 ↔ ¬ 𝑥 E 𝑦) |
9 | 5, 8 | bitr2i 279 | . . . . . . 7 ⊢ (¬ 𝑥 E 𝑦 ↔ 𝑦◡(V ∖ E )𝑥) |
10 | 9 | con1bii 360 | . . . . . 6 ⊢ (¬ 𝑦◡(V ∖ E )𝑥 ↔ 𝑥 E 𝑦) |
11 | epel 5433 | . . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
12 | 10, 11 | bitr2i 279 | . . . . 5 ⊢ (𝑥 ∈ 𝑦 ↔ ¬ 𝑦◡(V ∖ E )𝑥) |
13 | 12 | ralbii 3133 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡(V ∖ E )𝑥) |
14 | eldif 3891 | . . . . . 6 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (◡(V ∖ E ) “ 𝐴))) | |
15 | 4, 14 | mpbiran 708 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ¬ 𝑥 ∈ (◡(V ∖ E ) “ 𝐴)) |
16 | 4 | elima 5901 | . . . . 5 ⊢ (𝑥 ∈ (◡(V ∖ E ) “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥) |
17 | 15, 16 | xchbinx 337 | . . . 4 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ¬ ∃𝑦 ∈ 𝐴 𝑦◡(V ∖ E )𝑥) |
18 | 2, 13, 17 | 3bitr4ri 307 | . . 3 ⊢ (𝑥 ∈ (V ∖ (◡(V ∖ E ) “ 𝐴)) ↔ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
19 | 18 | abbi2i 2929 | . 2 ⊢ (V ∖ (◡(V ∖ E ) “ 𝐴)) = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} |
20 | 1, 19 | eqtr4i 2824 | 1 ⊢ ∩ 𝐴 = (V ∖ (◡(V ∖ E ) “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 {cab 2776 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ∖ cdif 3878 ∩ cint 4838 class class class wbr 5030 E cep 5429 ◡ccnv 5518 “ cima 5522 |
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-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 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-br 5031 df-opab 5093 df-eprel 5430 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: (None) |
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