Theorem lexoweon 35998

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 10117 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 35989	. 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 10117	1 ⊢ LexOrd We (On × On)

This proof depends on syntax axioms:   We wwe 5607   × cxp 5653  Oncon0 6380  LexOrdclexo 35980

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-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

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-rab 3417  df-v 3456  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-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-iota 6512  df-fun 6560  df-fv 6566  df-1st 8025  df-2nd 8026  df-lexo 35989

This theorem is used by: (None)