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Theorem djudom1 9602
Description: Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 1-Sep-2023.)
Assertion
Ref Expression
djudom1 ((𝐴𝐵𝐶𝑉) → (𝐴𝐶) ≼ (𝐵𝐶))

Proof of Theorem djudom1
StepHypRef Expression
1 snex 5328 . . . 4 {∅} ∈ V
21xpdom2 8606 . . 3 (𝐴𝐵 → ({∅} × 𝐴) ≼ ({∅} × 𝐵))
3 snex 5328 . . . . 5 {1o} ∈ V
4 xpexg 7466 . . . . 5 (({1o} ∈ V ∧ 𝐶𝑉) → ({1o} × 𝐶) ∈ V)
53, 4mpan 686 . . . 4 (𝐶𝑉 → ({1o} × 𝐶) ∈ V)
6 domrefg 8538 . . . 4 (({1o} × 𝐶) ∈ V → ({1o} × 𝐶) ≼ ({1o} × 𝐶))
75, 6syl 17 . . 3 (𝐶𝑉 → ({1o} × 𝐶) ≼ ({1o} × 𝐶))
8 xp01disjl 8117 . . . 4 (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅
9 undom 8599 . . . 4 (((({∅} × 𝐴) ≼ ({∅} × 𝐵) ∧ ({1o} × 𝐶) ≼ ({1o} × 𝐶)) ∧ (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅) → (({∅} × 𝐴) ∪ ({1o} × 𝐶)) ≼ (({∅} × 𝐵) ∪ ({1o} × 𝐶)))
108, 9mpan2 687 . . 3 ((({∅} × 𝐴) ≼ ({∅} × 𝐵) ∧ ({1o} × 𝐶) ≼ ({1o} × 𝐶)) → (({∅} × 𝐴) ∪ ({1o} × 𝐶)) ≼ (({∅} × 𝐵) ∪ ({1o} × 𝐶)))
112, 7, 10syl2an 595 . 2 ((𝐴𝐵𝐶𝑉) → (({∅} × 𝐴) ∪ ({1o} × 𝐶)) ≼ (({∅} × 𝐵) ∪ ({1o} × 𝐶)))
12 df-dju 9324 . 2 (𝐴𝐶) = (({∅} × 𝐴) ∪ ({1o} × 𝐶))
13 df-dju 9324 . 2 (𝐵𝐶) = (({∅} × 𝐵) ∪ ({1o} × 𝐶))
1411, 12, 133brtr4g 5097 1 ((𝐴𝐵𝐶𝑉) → (𝐴𝐶) ≼ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  Vcvv 3500  cun 3938  cin 3939  c0 4295  {csn 4564   class class class wbr 5063   × cxp 5552  1oc1o 8091  cdom 8501  cdju 9321
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-1o 8098  df-en 8504  df-dom 8505  df-dju 9324
This theorem is referenced by:  djudom2  9603  djulepw  9612  unctb  9621  infdif  9625  gchdjuidm  10084  gchpwdom  10086  gchhar  10095  pr2dom  39777  tr3dom  39778
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