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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 5576 | . . . . . . 7 ⊢ (𝑧 E 𝑥 ↔ 𝑧 ∈ 𝑥) | |
2 | brvdif 37640 | . . . . . . . 8 ⊢ (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 E 𝑦) | |
3 | epel 5576 | . . . . . . . 8 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦) | |
4 | 2, 3 | xchbinx 334 | . . . . . . 7 ⊢ (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 ∈ 𝑦) |
5 | 1, 4 | anbi12i 626 | . . . . . 6 ⊢ ((𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)) |
6 | 5 | exbii 1842 | . . . . 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 3966 | . . . 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 5208 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦} |
11 | rnxrn 37779 | . . . 4 ⊢ ran ( E ⋉ (V ∖ E )) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)} | |
12 | 11 | difeq2i 4114 | . . 3 ⊢ ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = ((V × V) ∖ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)}) |
13 | vvdifopab 37639 | . . 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 2754 | . 2 ⊢ ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = {⟨𝑥, 𝑦⟩ ∣ ¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)} |
15 | df-ssr 37879 | . 2 ⊢ S = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊆ 𝑦} | |
16 | 10, 14, 15 | 3eqtr4ri 2765 | 1 ⊢ S = ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 Vcvv 3468 ∖ cdif 3940 ⊆ wss 3943 class class class wbr 5141 {copab 5203 E cep 5572 × cxp 5667 ran crn 5670 ⋉ cxrn 37553 S cssr 37557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-eprel 5573 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fo 6542 df-fv 6544 df-1st 7971 df-2nd 7972 df-ec 8704 df-xrn 37752 df-ssr 37879 |
This theorem is referenced by: (None) |
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