MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dju1p1e2 Structured version   Visualization version   GIF version

Theorem dju1p1e2 10214
Description: 1+1=2 for cardinal number addition, derived from pm54.43 10041 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 9935), 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 10215 for a shorter proof that doesn't use pm54.43 10041. (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 9941 . 2 (1o ⊔ 1o) = (({∅} × 1o) ∪ ({1o} × 1o))
2 xp01disjl 8530 . . 3 (({∅} × 1o) ∩ ({1o} × 1o)) = ∅
3 0ex 5307 . . . . 5 ∅ ∈ V
4 1on 8518 . . . . 5 1o ∈ On
5 xpsnen2g 9105 . . . . 5 ((∅ ∈ V ∧ 1o ∈ On) → ({∅} × 1o) ≈ 1o)
63, 4, 5mp2an 692 . . . 4 ({∅} × 1o) ≈ 1o
7 xpsnen2g 9105 . . . . 5 ((1o ∈ On ∧ 1o ∈ On) → ({1o} × 1o) ≈ 1o)
84, 4, 7mp2an 692 . . . 4 ({1o} × 1o) ≈ 1o
9 pm54.43 10041 . . . 4 ((({∅} × 1o) ≈ 1o ∧ ({1o} × 1o) ≈ 1o) → ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o))
106, 8, 9mp2an 692 . . 3 ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o)
112, 10mpbi 230 . 2 (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o
121, 11eqbrtri 5164 1 (1o ⊔ 1o) ≈ 2o
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  cin 3950  c0 4333  {csn 4626   class class class wbr 5143   × cxp 5683  Oncon0 6384  1oc1o 8499  2oc2o 8500  cen 8982  cdju 9938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-dju 9941
This theorem is referenced by:  pr2dom  43540
  Copyright terms: Public domain W3C validator