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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfssr2 | Structured version Visualization version GIF version |
Description: Alternate definition of the subset relation. (Contributed by Peter Mazsa, 9-Aug-2021.) |
Ref | Expression |
---|---|
dfssr2 | ⊢ S = ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | epel 5585 | . . . . . . 7 ⊢ (𝑧 E 𝑥 ↔ 𝑧 ∈ 𝑥) | |
2 | brvdif 37733 | . . . . . . . 8 ⊢ (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 E 𝑦) | |
3 | epel 5585 | . . . . . . . 8 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦) | |
4 | 2, 3 | xchbinx 334 | . . . . . . 7 ⊢ (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 ∈ 𝑦) |
5 | 1, 4 | anbi12i 627 | . . . . . 6 ⊢ ((𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)) |
6 | 5 | exbii 1843 | . . . . 5 ⊢ (∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)) |
7 | 6 | notbii 320 | . . . 4 ⊢ (¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ ¬ ∃𝑧(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)) |
8 | dfss6 3969 | . . . 4 ⊢ (𝑥 ⊆ 𝑦 ↔ ¬ ∃𝑧(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)) | |
9 | 7, 8 | bitr4i 278 | . . 3 ⊢ (¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ 𝑥 ⊆ 𝑦) |
10 | 9 | opabbii 5215 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦} |
11 | rnxrn 37870 | . . . 4 ⊢ ran ( E ⋉ (V ∖ E )) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)} | |
12 | 11 | difeq2i 4117 | . . 3 ⊢ ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)}) |
13 | vvdifopab 37732 | . . 3 ⊢ ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)} | |
14 | 12, 13 | eqtri 2756 | . 2 ⊢ ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)} |
15 | df-ssr 37970 | . 2 ⊢ S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦} | |
16 | 10, 14, 15 | 3eqtr4ri 2767 | 1 ⊢ S = ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 Vcvv 3471 ∖ cdif 3944 ⊆ wss 3947 class class class wbr 5148 {copab 5210 E cep 5581 × cxp 5676 ran crn 5679 ⋉ cxrn 37647 S cssr 37651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-eprel 5582 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fo 6554 df-fv 6556 df-1st 7993 df-2nd 7994 df-ec 8727 df-xrn 37843 df-ssr 37970 |
This theorem is referenced by: (None) |
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