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Theorem wexp 8153
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 5678 . . 3 (𝑅 We 𝐴𝑅 Fr 𝐴)
2 wefr 5678 . . 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 8149 . . 3 ((𝑅 Fr 𝐴𝑆 Fr 𝐵) → 𝑇 Fr (𝐴 × 𝐵))
51, 2, 4syl2an 596 . 2 ((𝑅 We 𝐴𝑆 We 𝐵) → 𝑇 Fr (𝐴 × 𝐵))
6 weso 5679 . . 3 (𝑅 We 𝐴𝑅 Or 𝐴)
7 weso 5679 . . 3 (𝑆 We 𝐵𝑆 Or 𝐵)
83soxp 8152 . . 3 ((𝑅 Or 𝐴𝑆 Or 𝐵) → 𝑇 Or (𝐴 × 𝐵))
96, 7, 8syl2an 596 . 2 ((𝑅 We 𝐴𝑆 We 𝐵) → 𝑇 Or (𝐴 × 𝐵))
10 df-we 5642 . 2 (𝑇 We (𝐴 × 𝐵) ↔ (𝑇 Fr (𝐴 × 𝐵) ∧ 𝑇 Or (𝐴 × 𝐵)))
115, 9, 10sylanbrc 583 1 ((𝑅 We 𝐴𝑆 We 𝐵) → 𝑇 We (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1536  wcel 2105   class class class wbr 5147  {copab 5209   Or wor 5595   Fr wfr 5637   We wwe 5639   × cxp 5686  cfv 6562  1st c1st 8010  2nd c2nd 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fv 6570  df-1st 8012  df-2nd 8013
This theorem is referenced by:  fnwelem  8154  leweon  10048
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