![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > noxpordpo | Structured version Visualization version GIF version |
Description: To get through most of the textbook defintions 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.) |
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 |
---|---|
noxpordpo | ⊢ 𝑆 Po ( 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 | lrrecpo 27256 | . . . 4 ⊢ 𝑅 Po No |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → 𝑅 Po No ) |
5 | 1, 4, 4 | poxp2 8076 | . 2 ⊢ (⊤ → 𝑆 Po ( No × No )) |
6 | 5 | mptru 1549 | 1 ⊢ 𝑆 Po ( No × No ) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 846 ∧ w3a 1088 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ≠ wne 2944 ∪ cun 3909 class class class wbr 5106 {copab 5168 Po wpo 5544 × cxp 5632 ‘cfv 6497 1st c1st 7920 2nd c2nd 7921 No csur 26991 L cleft 27178 R cright 27179 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-1o 8413 df-2o 8414 df-no 26994 df-slt 26995 df-bday 26996 df-sslt 27124 df-scut 27126 df-made 27180 df-old 27181 df-left 27183 df-right 27184 |
This theorem is referenced by: norec2fn 27271 norec2ov 27272 |
Copyright terms: Public domain | W3C validator |