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Theorem wexp 7236
Description: A lexicographical ordering of two well-ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
Hypothesis
Ref Expression
wexp.1 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))))}
Assertion
Ref Expression
wexp ((𝑅 We 𝐴𝑆 We 𝐵) → 𝑇 We (𝐴 × 𝐵))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝑇(𝑥,𝑦)

Proof of Theorem wexp
StepHypRef Expression
1 wefr 5064 . . 3 (𝑅 We 𝐴𝑅 Fr 𝐴)
2 wefr 5064 . . 3 (𝑆 We 𝐵𝑆 Fr 𝐵)
3 wexp.1 . . . 4 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st𝑥)𝑅(1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥)𝑆(2nd𝑦))))}
43frxp 7232 . . 3 ((𝑅 Fr 𝐴𝑆 Fr 𝐵) → 𝑇 Fr (𝐴 × 𝐵))
51, 2, 4syl2an 494 . 2 ((𝑅 We 𝐴𝑆 We 𝐵) → 𝑇 Fr (𝐴 × 𝐵))
6 weso 5065 . . 3 (𝑅 We 𝐴𝑅 Or 𝐴)
7 weso 5065 . . 3 (𝑆 We 𝐵𝑆 Or 𝐵)
83soxp 7235 . . 3 ((𝑅 Or 𝐴𝑆 Or 𝐵) → 𝑇 Or (𝐴 × 𝐵))
96, 7, 8syl2an 494 . 2 ((𝑅 We 𝐴𝑆 We 𝐵) → 𝑇 Or (𝐴 × 𝐵))
10 df-we 5035 . 2 (𝑇 We (𝐴 × 𝐵) ↔ (𝑇 Fr (𝐴 × 𝐵) ∧ 𝑇 Or (𝐴 × 𝐵)))
115, 9, 10sylanbrc 697 1 ((𝑅 We 𝐴𝑆 We 𝐵) → 𝑇 We (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384   = wceq 1480  wcel 1987   class class class wbr 4613  {copab 4672   Or wor 4994   Fr wfr 5030   We wwe 5032   × cxp 5072  cfv 5847  1st c1st 7111  2nd c2nd 7112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-1st 7113  df-2nd 7114
This theorem is referenced by:  fnwelem  7237  leweon  8778
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