noxpordse - Metamath Proof Explorer

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Theorem noxpordse 28350

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 28340	. . . 4 ⊢ 𝑅 Se No
4	3	a1i 11	. . 3 ⊢ (⊤ → 𝑅 Se No)
5	1, 4, 4	sexp2 8173	. 2 ⊢ (⊤ → 𝑆 Se (No × No))
6	5	mptru 1576	1 ⊢ 𝑆 Se (No × No)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∨ wo 861 ∧ w3a 1103 = wceq 1569 ⊤wtru 1570 ∈ wcel 2146 ≠ wne 2958 ∪ cun 4151 class class class wbr 5105 {copab 5169 Se wse 5605 × cxp 5652 ‘cfv 6546 1^st c1st 8006 2^nd c2nd 8007 Nocsur 28009 Lcleft 28223 Rcright 28224

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3416 df-v 3455 df-ss 3900 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-br 5106 df-opab 5170 df-mpt 5189 df-id 5549 df-se 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-iota 6500 df-fun 6548 df-fv 6554 df-1st 8008 df-2nd 8009

This theorem is used by: norec2fn 28354 norec2ov 28355

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