Theorem dju1p1e2 10088

Description: 1+1=2 for cardinal number addition, derived from pm54.43 9917 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 9811), 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 10089 for a shorter proof that doesn't use pm54.43 9917. (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 9817	. 2 ⊢ (1o ⊔ 1o) = (({∅} × 1o) ∪ ({1o} × 1o))
2		xp01disjl 8421	. . 3 ⊢ (({∅} × 1o) ∩ ({1o} × 1o)) = ∅
3		0ex 5253	. . . . 5 ⊢ ∅ ∈ V
4		1on 8411	. . . . 5 ⊢ 1o ∈ On
5		xpsnen2g 9002	. . . . 5 ⊢ ((∅ ∈ V ∧ 1o ∈ On) → ({∅} × 1o) ≈ 1o)
6	3, 4, 5	mp2an 693	. . . 4 ⊢ ({∅} × 1o) ≈ 1o
7		xpsnen2g 9002	. . . . 5 ⊢ ((1o ∈ On ∧ 1o ∈ On) → ({1o} × 1o) ≈ 1o)
8	4, 4, 7	mp2an 693	. . . 4 ⊢ ({1o} × 1o) ≈ 1o
9		pm54.43 9917	. . . 4 ⊢ ((({∅} × 1o) ≈ 1o ∧ ({1o} × 1o) ≈ 1o) → ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o))
10	6, 8, 9	mp2an 693	. . 3 ⊢ ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o)
11	2, 10	mpbi 230	. 2 ⊢ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o
12	1, 11	eqbrtri 5120	1 ⊢ (1o ⊔ 1o) ≈ 2o

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  Vcvv 3441   ∪ cun 3900   ∩ cin 3901  ∅c0 4286  {csn 4581   class class class wbr 5099   × cxp 5623  Oncon0 6318  1_oc1o 8392  2_oc2o 8393   ≈ cen 8884   ⊔ cdju 9814

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  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 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-1st 7935  df-2nd 7936  df-1o 8399  df-2o 8400  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-dju 9817

This theorem is referenced by:  pr2dom  43835