Theorem dju1p1e2 10035

Description: 1+1=2 for cardinal number addition, derived from pm54.43 9863 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 9757), 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 10036 for a shorter proof that doesn't use pm54.43 9863. (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 9763	. 2 ⊢ (1o ⊔ 1o) = (({∅} × 1o) ∪ ({1o} × 1o))
2		xp01disjl 8398	. . 3 ⊢ (({∅} × 1o) ∩ ({1o} × 1o)) = ∅
3		0ex 5256	. . . . 5 ⊢ ∅ ∈ V
4		1on 8384	. . . . 5 ⊢ 1o ∈ On
5		xpsnen2g 8935	. . . . 5 ⊢ ((∅ ∈ V ∧ 1o ∈ On) → ({∅} × 1o) ≈ 1o)
6	3, 4, 5	mp2an 690	. . . 4 ⊢ ({∅} × 1o) ≈ 1o
7		xpsnen2g 8935	. . . . 5 ⊢ ((1o ∈ On ∧ 1o ∈ On) → ({1o} × 1o) ≈ 1o)
8	4, 4, 7	mp2an 690	. . . 4 ⊢ ({1o} × 1o) ≈ 1o
9		pm54.43 9863	. . . 4 ⊢ ((({∅} × 1o) ≈ 1o ∧ ({1o} × 1o) ≈ 1o) → ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o))
10	6, 8, 9	mp2an 690	. . 3 ⊢ ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o)
11	2, 10	mpbi 229	. 2 ⊢ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o
12	1, 11	eqbrtri 5118	1 ⊢ (1o ⊔ 1o) ≈ 2o

Syntax hints:   ↔ wb 205   = wceq 1541   ∈ wcel 2106  Vcvv 3442   ∪ cun 3900   ∩ cin 3901  ∅c0 4274  {csn 4578   class class class wbr 5097   × cxp 5623  Oncon0 6307  1_oc1o 8365  2_oc2o 8366   ≈ cen 8806   ⊔ cdju 9760

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-int 4900  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6310  df-on 6311  df-suc 6313  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-1st 7904  df-2nd 7905  df-1o 8372  df-2o 8373  df-er 8574  df-en 8810  df-dom 8811  df-sdom 8812  df-dju 9763

This theorem is referenced by:  pr2dom  41506