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Theorem wexp 7440
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 5239 . . 3 (𝑅 We 𝐴𝑅 Fr 𝐴)
2 wefr 5239 . . 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 7436 . . 3 ((𝑅 Fr 𝐴𝑆 Fr 𝐵) → 𝑇 Fr (𝐴 × 𝐵))
51, 2, 4syl2an 583 . 2 ((𝑅 We 𝐴𝑆 We 𝐵) → 𝑇 Fr (𝐴 × 𝐵))
6 weso 5240 . . 3 (𝑅 We 𝐴𝑅 Or 𝐴)
7 weso 5240 . . 3 (𝑆 We 𝐵𝑆 Or 𝐵)
83soxp 7439 . . 3 ((𝑅 Or 𝐴𝑆 Or 𝐵) → 𝑇 Or (𝐴 × 𝐵))
96, 7, 8syl2an 583 . 2 ((𝑅 We 𝐴𝑆 We 𝐵) → 𝑇 Or (𝐴 × 𝐵))
10 df-we 5210 . 2 (𝑇 We (𝐴 × 𝐵) ↔ (𝑇 Fr (𝐴 × 𝐵) ∧ 𝑇 Or (𝐴 × 𝐵)))
115, 9, 10sylanbrc 572 1 ((𝑅 We 𝐴𝑆 We 𝐵) → 𝑇 We (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 836   = wceq 1631  wcel 2145   class class class wbr 4786  {copab 4846   Or wor 5169   Fr wfr 5205   We wwe 5207   × cxp 5247  cfv 6029  1st c1st 7311  2nd c2nd 7312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fv 6037  df-1st 7313  df-2nd 7314
This theorem is referenced by:  fnwelem  7441  leweon  9032
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