noxpordse - Mathbox for Scott Fenton

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Theorem noxpordse 34117

Description: Next we establish the set-like nature of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)

**Hypotheses**
Ref	Expression
noxpord.1	⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}
noxpord.2	⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1^st ‘𝑥)𝑅(1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥)𝑅(2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}

**Assertion**
Ref	Expression
noxpordse	⊢ 𝑆 Se ( No × No )

Distinct variable groups: 𝑥,𝑅,𝑦 𝑎,𝑏 Allowed substitution hints: 𝑅(𝑎,𝑏) 𝑆(𝑥,𝑦,𝑎,𝑏)

**Proof of Theorem noxpordse**
Step	Hyp	Ref	Expression
1		noxpord.2	. . 3 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1^st ‘𝑥)𝑅(1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥)𝑅(2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}
2		noxpord.1	. . . . 5 ⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}
3	2	lrrecse 34107	. . . 4 ⊢ 𝑅 Se No
4	3	a1i 11	. . 3 ⊢ (⊤ → 𝑅 Se No )
5	1, 4, 4	sexp2 33801	. 2 ⊢ (⊤ → 𝑆 Se ( No × No ))
6	5	mptru 1546	1 ⊢ 𝑆 Se ( No × No )

Colors of variables: wff setvar class

Syntax hints: ∨ wo 844 ∧ w3a 1086 = wceq 1539 ⊤wtru 1540 ∈ wcel 2106 ≠ wne 2943 ∪ cun 3884 class class class wbr 5073 {copab 5135 Se wse 5537 × cxp 5582 ‘cfv 6426 1^st c1st 7818 2^nd c2nd 7819 No csur 33851 L cleft 34037 R cright 34038

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-se 5540 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-iota 6384 df-fun 6428 df-fv 6434 df-1st 7820 df-2nd 7821

This theorem is referenced by: norec2fn 34121 norec2ov 34122

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