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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 5433 | . . . . . . 7 ⊢ (𝑧 E 𝑥 ↔ 𝑧 ∈ 𝑥) | |
2 | brvdif 35682 | . . . . . . . 8 ⊢ (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 E 𝑦) | |
3 | epel 5433 | . . . . . . . 8 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦) | |
4 | 2, 3 | xchbinx 337 | . . . . . . 7 ⊢ (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 ∈ 𝑦) |
5 | 1, 4 | anbi12i 629 | . . . . . 6 ⊢ ((𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)) |
6 | 5 | exbii 1849 | . . . . 5 ⊢ (∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)) |
7 | 6 | notbii 323 | . . . 4 ⊢ (¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ ¬ ∃𝑧(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)) |
8 | dfss6 3904 | . . . 4 ⊢ (𝑥 ⊆ 𝑦 ↔ ¬ ∃𝑧(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)) | |
9 | 7, 8 | bitr4i 281 | . . 3 ⊢ (¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ 𝑥 ⊆ 𝑦) |
10 | 9 | opabbii 5097 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)} = {〈𝑥, 𝑦〉 ∣ 𝑥 ⊆ 𝑦} |
11 | rnxrn 35806 | . . . 4 ⊢ ran ( E ⋉ (V ∖ E )) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)} | |
12 | 11 | difeq2i 4047 | . . 3 ⊢ ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = ((V × V) ∖ {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)}) |
13 | vvdifopab 35681 | . . 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 2821 | . 2 ⊢ ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = {〈𝑥, 𝑦〉 ∣ ¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)} |
15 | df-ssr 35898 | . 2 ⊢ S = {〈𝑥, 𝑦〉 ∣ 𝑥 ⊆ 𝑦} | |
16 | 10, 14, 15 | 3eqtr4ri 2832 | 1 ⊢ S = ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 class class class wbr 5030 {copab 5092 E cep 5429 × cxp 5517 ran crn 5520 ⋉ cxrn 35612 S cssr 35616 |
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-pow 5231 ax-pr 5295 ax-un 7441 |
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-rab 3115 df-v 3443 df-sbc 3721 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-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-eprel 5430 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-1st 7671 df-2nd 7672 df-ec 8274 df-xrn 35783 df-ssr 35898 |
This theorem is referenced by: (None) |
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