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Theorem leweon 9650
Description: Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 9651, 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 7579 . 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 6749 . . . . . . . 8 (1st𝑦) ∈ V
43epeli 5477 . . . . . . 7 ((1st𝑥) E (1st𝑦) ↔ (1st𝑥) ∈ (1st𝑦))
5 fvex 6749 . . . . . . . . 9 (2nd𝑦) ∈ V
65epeli 5477 . . . . . . . 8 ((2nd𝑥) E (2nd𝑦) ↔ (2nd𝑥) ∈ (2nd𝑦))
76anbi2i 626 . . . . . . 7 (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦)) ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦)))
84, 7orbi12i 915 . . . . . 6 (((1st𝑥) E (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦))) ↔ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))
98anbi2i 626 . . . . 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 5135 . . . 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 2769 . . 3 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) E (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦))))}
1211wexp 7918 . 2 (( E We On ∧ E We On) → 𝐿 We (On × On))
131, 1, 12mp2an 692 1 𝐿 We (On × On)
Colors of variables: wff setvar class
Syntax hints:  wa 399  wo 847   = wceq 1543  wcel 2111   class class class wbr 5068  {copab 5130   E cep 5474   We wwe 5523   × cxp 5564  Oncon0 6231  cfv 6398  1st c1st 7778  2nd c2nd 7779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337  ax-un 7542
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4835  df-int 4875  df-br 5069  df-opab 5131  df-mpt 5151  df-tr 5177  df-id 5470  df-eprel 5475  df-po 5483  df-so 5484  df-fr 5524  df-we 5526  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-ord 6234  df-on 6235  df-iota 6356  df-fun 6400  df-fv 6406  df-1st 7780  df-2nd 7781
This theorem is referenced by:  r0weon  9651
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