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Mirrors > Home > MPE Home > Th. List > Mathboxes > noxpordse | Structured version Visualization version GIF version |
Description: Next we establish the set-like nature of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.) |
Ref | Expression |
---|---|
noxpord.1 | ⊢ 𝑅 = {〈𝑎, 𝑏〉 ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} |
noxpord.2 | ⊢ 𝑆 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} |
Ref | Expression |
---|---|
noxpordse | ⊢ 𝑆 Se ( No × No ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noxpord.2 | . . 3 ⊢ 𝑆 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} | |
2 | noxpord.1 | . . . . 5 ⊢ 𝑅 = {〈𝑎, 𝑏〉 ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))} | |
3 | 2 | lrrecse 33681 | . . . 4 ⊢ 𝑅 Se No |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → 𝑅 Se No ) |
5 | 1, 4, 4 | sexp2 33360 | . 2 ⊢ (⊤ → 𝑆 Se ( No × No )) |
6 | 5 | mptru 1545 | 1 ⊢ 𝑆 Se ( No × No ) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 844 ∧ w3a 1084 = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 ≠ wne 2951 ∪ cun 3858 class class class wbr 5036 {copab 5098 Se wse 5485 × cxp 5526 ‘cfv 6340 1st c1st 7697 2nd c2nd 7698 No csur 33440 L cleft 33623 R cright 33624 |
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 2729 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3699 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-br 5037 df-opab 5099 df-mpt 5117 df-id 5434 df-se 5488 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-iota 6299 df-fun 6342 df-fv 6348 df-1st 7699 df-2nd 7700 |
This theorem is referenced by: norec2fn 33695 norec2ov 33696 |
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