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

Theorem leweon 9124
 Description: Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 9125, 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 7220 . 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 6428 . . . . . . . 8 (1st𝑦) ∈ V
43epeli 5231 . . . . . . 7 ((1st𝑥) E (1st𝑦) ↔ (1st𝑥) ∈ (1st𝑦))
5 fvex 6428 . . . . . . . . 9 (2nd𝑦) ∈ V
65epeli 5231 . . . . . . . 8 ((2nd𝑥) E (2nd𝑦) ↔ (2nd𝑥) ∈ (2nd𝑦))
76anbi2i 617 . . . . . . 7 (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦)) ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦)))
84, 7orbi12i 939 . . . . . 6 (((1st𝑥) E (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦))) ↔ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))
98anbi2i 617 . . . . 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 4914 . . . 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 2828 . . 3 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) E (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) E (2nd𝑦))))}
1211wexp 7532 . 2 (( E We On ∧ E We On) → 𝐿 We (On × On))
131, 1, 12mp2an 684 1 𝐿 We (On × On)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 385   ∨ wo 874   = wceq 1653   ∈ wcel 2157   class class class wbr 4847  {copab 4909   E cep 5228   We wwe 5274   × cxp 5314  Oncon0 5945  ‘cfv 6105  1st c1st 7403  2nd c2nd 7404 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101  ax-un 7187 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ne 2976  df-ral 3098  df-rex 3099  df-rab 3102  df-v 3391  df-sbc 3638  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-pss 3789  df-nul 4120  df-if 4282  df-sn 4373  df-pr 4375  df-tp 4377  df-op 4379  df-uni 4633  df-int 4672  df-br 4848  df-opab 4910  df-mpt 4927  df-tr 4950  df-id 5224  df-eprel 5229  df-po 5237  df-so 5238  df-fr 5275  df-we 5277  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-rn 5327  df-res 5328  df-ima 5329  df-ord 5948  df-on 5949  df-iota 6068  df-fun 6107  df-fv 6113  df-1st 7405  df-2nd 7406 This theorem is referenced by:  r0weon  9125
 Copyright terms: Public domain W3C validator