Theorem lexoweon 35930

Description: Lexicographical order is a well-ordering of On × On. Part of Proposition 7.56(1) of [TakeutiZaring] p. 54. Unlike 𝑅₀, this order is not set-like, as the preimage of ⟨1_o, ∅⟩ is the proper class ({∅} × On). This theorem is leweon 10091 expressed using LexOrd. (Contributed by BTernaryTau, 19-Sep-2026.)

Assertion

Ref	Expression
lexoweon	⊢ LexOrd We (On × On)

Proof of Theorem lexoweon

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lexo 35921	. 2 ⊢ LexOrd = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))))}
2	1	leweon 10091	1 ⊢ LexOrd We (On × On)

This proof depends on syntax axioms:   We wwe 5602   × cxp 5648  Oncon0 6362  LexOrdclexo 35912

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-iota 6494  df-fun 6540  df-fv 6546  df-1st 8001  df-2nd 8002  df-lexo 35921

This theorem is used by: (None)