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Theorem leweon 8694
Description: Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 8695, this order is not set-like, as the preimage of ⟨1𝑜, ∅⟩ is the proper class ({∅} × On). (Contributed by Mario Carneiro, 9-Mar-2013.)
Hypothesis
Ref Expression
leweon.1 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}
Assertion
Ref Expression
leweon 𝐿 We (On × On)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐿(𝑥,𝑦)

Proof of Theorem leweon
StepHypRef Expression
1 epweon 6852 . 2 E We On
2 leweon.1 . . . 4 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}
3 fvex 6098 . . . . . . . 8 (1st𝑦) ∈ V
43epelc 4941 . . . . . . 7 ((1st𝑥) E (1st𝑦) ↔ (1st𝑥) ∈ (1st𝑦))
5 fvex 6098 . . . . . . . . 9 (2nd𝑦) ∈ V
65epelc 4941 . . . . . . . 8 ((2nd𝑥) E (2nd𝑦) ↔ (2nd𝑥) ∈ (2nd𝑦))
76anbi2i 725 . . . . . . 7 (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦)) ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦)))
84, 7orbi12i 541 . . . . . 6 (((1st𝑥) E (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦))) ↔ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))
98anbi2i 725 . . . . 5 (((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) E (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦)))) ↔ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦)))))
109opabbii 4643 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) E (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}
112, 10eqtr4i 2634 . . 3 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) E (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦))))}
1211wexp 7155 . 2 (( E We On ∧ E We On) → 𝐿 We (On × On))
131, 1, 12mp2an 703 1 𝐿 We (On × On)
Colors of variables: wff setvar class
Syntax hints:  wo 381  wa 382   = wceq 1474  wcel 1976   class class class wbr 4577  {copab 4636   E cep 4937   We wwe 4986   × cxp 5026  Oncon0 5626  cfv 5790  1st c1st 7034  2nd c2nd 7035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-iota 5754  df-fun 5792  df-fv 5798  df-1st 7036  df-2nd 7037
This theorem is referenced by:  r0weon  8695
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