Theorem dju1p1e2 10212

Description: 1+1=2 for cardinal number addition, derived from pm54.43 10039 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 9933), 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 10213 for a shorter proof that doesn't use pm54.43 10039. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)

Assertion

Ref	Expression
dju1p1e2	⊢ (1o ⊔ 1o) ≈ 2o

Proof of Theorem dju1p1e2

Step	Hyp	Ref	Expression
1		df-dju 9939	. 2 ⊢ (1o ⊔ 1o) = (({∅} × 1o) ∪ ({1o} × 1o))
2		xp01disjl 8529	. . 3 ⊢ (({∅} × 1o) ∩ ({1o} × 1o)) = ∅
3		0ex 5313	. . . . 5 ⊢ ∅ ∈ V
4		1on 8517	. . . . 5 ⊢ 1o ∈ On
5		xpsnen2g 9104	. . . . 5 ⊢ ((∅ ∈ V ∧ 1o ∈ On) → ({∅} × 1o) ≈ 1o)
6	3, 4, 5	mp2an 692	. . . 4 ⊢ ({∅} × 1o) ≈ 1o
7		xpsnen2g 9104	. . . . 5 ⊢ ((1o ∈ On ∧ 1o ∈ On) → ({1o} × 1o) ≈ 1o)
8	4, 4, 7	mp2an 692	. . . 4 ⊢ ({1o} × 1o) ≈ 1o
9		pm54.43 10039	. . . 4 ⊢ ((({∅} × 1o) ≈ 1o ∧ ({1o} × 1o) ≈ 1o) → ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o))
10	6, 8, 9	mp2an 692	. . 3 ⊢ ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o)
11	2, 10	mpbi 230	. 2 ⊢ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o
12	1, 11	eqbrtri 5169	1 ⊢ (1o ⊔ 1o) ≈ 2o

Syntax hints:   ↔ wb 206   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∪ cun 3961   ∩ cin 3962  ∅c0 4339  {csn 4631   class class class wbr 5148   × cxp 5687  Oncon0 6386  1_oc1o 8498  2_oc2o 8499   ≈ cen 8981   ⊔ cdju 9936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-dju 9939

This theorem is referenced by:  pr2dom  43517