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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 5462 | . . . . . . 7 ⊢ (𝑧 E 𝑥 ↔ 𝑧 ∈ 𝑥) | |
2 | brvdif 35403 | . . . . . . . 8 ⊢ (𝑧(V ∖ E )𝑦 ↔ ¬ 𝑧 E 𝑦) | |
3 | epel 5462 | . . . . . . . 8 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦) | |
4 | 2, 3 | xchbinx 335 | . . . . . . 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 1839 | . . . . 5 ⊢ (∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)) |
7 | 6 | notbii 321 | . . . 4 ⊢ (¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ ¬ ∃𝑧(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)) |
8 | dfss6 3954 | . . . 4 ⊢ (𝑥 ⊆ 𝑦 ↔ ¬ ∃𝑧(𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)) | |
9 | 7, 8 | bitr4i 279 | . . 3 ⊢ (¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦) ↔ 𝑥 ⊆ 𝑦) |
10 | 9 | opabbii 5124 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)} = {〈𝑥, 𝑦〉 ∣ 𝑥 ⊆ 𝑦} |
11 | rnxrn 35526 | . . . 4 ⊢ ran ( E ⋉ (V ∖ E )) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)} | |
12 | 11 | difeq2i 4093 | . . 3 ⊢ ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = ((V × V) ∖ {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)}) |
13 | vvdifopab 35402 | . . 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 2841 | . 2 ⊢ ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) = {〈𝑥, 𝑦〉 ∣ ¬ ∃𝑧(𝑧 E 𝑥 ∧ 𝑧(V ∖ E )𝑦)} |
15 | df-ssr 35618 | . 2 ⊢ S = {〈𝑥, 𝑦〉 ∣ 𝑥 ⊆ 𝑦} | |
16 | 10, 14, 15 | 3eqtr4ri 2852 | 1 ⊢ S = ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 class class class wbr 5057 {copab 5119 E cep 5457 × cxp 5546 ran crn 5549 ⋉ cxrn 35333 S cssr 35337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-eprel 5458 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-1st 7678 df-2nd 7679 df-ec 8280 df-xrn 35503 df-ssr 35618 |
This theorem is referenced by: (None) |
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