noxpordpo - Metamath Proof Explorer

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Theorem noxpordpo 28393

Description: To get through most of the textbook definitions in surreal numbers we will need recursion on two variables. This set of theorems sets up the preconditions for double recursion. This theorem establishes the partial ordering. (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
noxpordpo	⊢ 𝑆 Po (No × No)

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

**Proof of Theorem noxpordpo**
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	lrrecpo 28384	. . . 4 ⊢ 𝑅 Po No
4	3	a1i 11	. . 3 ⊢ (⊤ → 𝑅 Po No)
5	1, 4, 4	poxp2 8197	. 2 ⊢ (⊤ → 𝑆 Po (No × No))
6	5	mptru 1577	1 ⊢ 𝑆 Po (No × No)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∨ wo 861 ∧ w3a 1103 = wceq 1570 ⊤wtru 1571 ∈ wcel 2147 ≠ wne 2959 ∪ cun 4152 class class class wbr 5108 {copab 5173 Po wpo 5564 × cxp 5656 ‘cfv 6566 1^st c1st 8033 2^nd c2nd 8034 Nocsur 28054 Lcleft 28268 Rcright 28269

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-1o 8514 df-2o 8515 df-no 28057 df-lts 28058 df-bday 28059 df-slts 28201 df-cuts 28203 df-made 28270 df-old 28271 df-left 28273 df-right 28274

This theorem is used by: norec2fn 28399 norec2ov 28400

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