noxpordse - Metamath Proof Explorer

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Theorem noxpordse 27436

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 27426	. . . 4 ⊢ 𝑅 Se No
4	3	a1i 11	. . 3 ⊢ (⊤ → 𝑅 Se No )
5	1, 4, 4	sexp2 8132	. 2 ⊢ (⊤ → 𝑆 Se ( No × No ))
6	5	mptru 1549	1 ⊢ 𝑆 Se ( No × No )

Colors of variables: wff setvar class

Syntax hints: ∨ wo 846 ∧ w3a 1088 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ≠ wne 2941 ∪ cun 3947 class class class wbr 5149 {copab 5211 Se wse 5630 × cxp 5675 ‘cfv 6544 1^st c1st 7973 2^nd c2nd 7974 No csur 27143 L cleft 27340 R cright 27341

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-se 5633 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-iota 6496 df-fun 6546 df-fv 6552 df-1st 7975 df-2nd 7976

This theorem is referenced by: norec2fn 27440 norec2ov 27441

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