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Theorem leweon 10035
Description: Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 10036, 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 7777 . 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 6910 . . . . . . . 8 (1st𝑦) ∈ V
43epeli 5584 . . . . . . 7 ((1st𝑥) E (1st𝑦) ↔ (1st𝑥) ∈ (1st𝑦))
5 fvex 6910 . . . . . . . . 9 (2nd𝑦) ∈ V
65epeli 5584 . . . . . . . 8 ((2nd𝑥) E (2nd𝑦) ↔ (2nd𝑥) ∈ (2nd𝑦))
76anbi2i 622 . . . . . . 7 (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦)) ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦)))
84, 7orbi12i 913 . . . . . 6 (((1st𝑥) E (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦))) ↔ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))
98anbi2i 622 . . . . 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 5215 . . . 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 2759 . . 3 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) E (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦))))}
1211wexp 8135 . 2 (( E We On ∧ E We On) → 𝐿 We (On × On))
131, 1, 12mp2an 691 1 𝐿 We (On × On)
Colors of variables: wff setvar class
Syntax hints:  wa 395  wo 846   = wceq 1534  wcel 2099   class class class wbr 5148  {copab 5210   E cep 5581   We wwe 5632   × cxp 5676  Oncon0 6369  cfv 6548  1st c1st 7991  2nd c2nd 7992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-iota 6500  df-fun 6550  df-fv 6556  df-1st 7993  df-2nd 7994
This theorem is referenced by:  r0weon  10036
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