MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  leweon Structured version   Visualization version   GIF version

Theorem leweon 10005
Description: Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 10006, 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 7758 . 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 6897 . . . . . . . 8 (1st𝑦) ∈ V
43epeli 5575 . . . . . . 7 ((1st𝑥) E (1st𝑦) ↔ (1st𝑥) ∈ (1st𝑦))
5 fvex 6897 . . . . . . . . 9 (2nd𝑦) ∈ V
65epeli 5575 . . . . . . . 8 ((2nd𝑥) E (2nd𝑦) ↔ (2nd𝑥) ∈ (2nd𝑦))
76anbi2i 622 . . . . . . 7 (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦)) ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦)))
84, 7orbi12i 911 . . . . . 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 5208 . . . 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 2757 . . 3 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) E (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦))))}
1211wexp 8113 . 2 (( E We On ∧ E We On) → 𝐿 We (On × On))
131, 1, 12mp2an 689 1 𝐿 We (On × On)
Colors of variables: wff setvar class
Syntax hints:  wa 395  wo 844   = wceq 1533  wcel 2098   class class class wbr 5141  {copab 5203   E cep 5572   We wwe 5623   × cxp 5667  Oncon0 6357  cfv 6536  1st c1st 7969  2nd c2nd 7970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-iota 6488  df-fun 6538  df-fv 6544  df-1st 7971  df-2nd 7972
This theorem is referenced by:  r0weon  10006
  Copyright terms: Public domain W3C validator