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Theorem dju1p1e2 9599
Description: 1+1=2 for cardinal number addition, derived from pm54.43 9429 as promised. Theorem *110.643 of Principia Mathematica, vol. II, p. 86, which adds the remark, "The above proposition is occasionally useful." Whitehead and Russell define cardinal addition on collections of all sets equinumerous to 1 and 2 (which for us are proper classes unless we restrict them as in karden 9323), but after applying definitions, our theorem is equivalent. Because we use a disjoint union for cardinal addition (as explained in the comment at the top of this section), we use instead of =. See dju1p1e2ALT 9600 for a shorter proof that doesn't use pm54.43 9429. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)
Assertion
Ref Expression
dju1p1e2 (1o ⊔ 1o) ≈ 2o

Proof of Theorem dju1p1e2
StepHypRef Expression
1 df-dju 9329 . 2 (1o ⊔ 1o) = (({∅} × 1o) ∪ ({1o} × 1o))
2 xp01disjl 8119 . . 3 (({∅} × 1o) ∩ ({1o} × 1o)) = ∅
3 0ex 5198 . . . . 5 ∅ ∈ V
4 1on 8107 . . . . 5 1o ∈ On
5 xpsnen2g 8608 . . . . 5 ((∅ ∈ V ∧ 1o ∈ On) → ({∅} × 1o) ≈ 1o)
63, 4, 5mp2an 691 . . . 4 ({∅} × 1o) ≈ 1o
7 xpsnen2g 8608 . . . . 5 ((1o ∈ On ∧ 1o ∈ On) → ({1o} × 1o) ≈ 1o)
84, 4, 7mp2an 691 . . . 4 ({1o} × 1o) ≈ 1o
9 pm54.43 9429 . . . 4 ((({∅} × 1o) ≈ 1o ∧ ({1o} × 1o) ≈ 1o) → ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o))
106, 8, 9mp2an 691 . . 3 ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o)
112, 10mpbi 233 . 2 (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o
121, 11eqbrtri 5074 1 (1o ⊔ 1o) ≈ 2o
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2115  Vcvv 3480  cun 3917  cin 3918  c0 4276  {csn 4550   class class class wbr 5053   × cxp 5541  Oncon0 6180  1oc1o 8093  2oc2o 8094  cen 8504  cdju 9326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-om 7577  df-1st 7686  df-2nd 7687  df-1o 8100  df-2o 8101  df-er 8287  df-en 8508  df-dom 8509  df-sdom 8510  df-dju 9329
This theorem is referenced by:  pr2dom  40179
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