Theorem dju1p1e2 9584

Description: 1+1=2 for cardinal number addition, derived from pm54.43 9414 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 9308), 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 9585 for a shorter proof that doesn't use pm54.43 9414. (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 9314	. 2 ⊢ (1o ⊔ 1o) = (({∅} × 1o) ∪ ({1o} × 1o))
2		xp01disjl 8104	. . 3 ⊢ (({∅} × 1o) ∩ ({1o} × 1o)) = ∅
3		0ex 5175	. . . . 5 ⊢ ∅ ∈ V
4		1on 8092	. . . . 5 ⊢ 1o ∈ On
5		xpsnen2g 8593	. . . . 5 ⊢ ((∅ ∈ V ∧ 1o ∈ On) → ({∅} × 1o) ≈ 1o)
6	3, 4, 5	mp2an 691	. . . 4 ⊢ ({∅} × 1o) ≈ 1o
7		xpsnen2g 8593	. . . . 5 ⊢ ((1o ∈ On ∧ 1o ∈ On) → ({1o} × 1o) ≈ 1o)
8	4, 4, 7	mp2an 691	. . . 4 ⊢ ({1o} × 1o) ≈ 1o
9		pm54.43 9414	. . . 4 ⊢ ((({∅} × 1o) ≈ 1o ∧ ({1o} × 1o) ≈ 1o) → ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o))
10	6, 8, 9	mp2an 691	. . 3 ⊢ ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o)
11	2, 10	mpbi 233	. 2 ⊢ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o
12	1, 11	eqbrtri 5051	1 ⊢ (1o ⊔ 1o) ≈ 2o

Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∪ cun 3879   ∩ cin 3880  ∅c0 4243  {csn 4525   class class class wbr 5030   × cxp 5517  Oncon0 6159  1_oc1o 8078  2_oc2o 8079   ≈ cen 8489   ⊔ cdju 9311

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-1st 7671  df-2nd 7672  df-1o 8085  df-2o 8086  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-dju 9314

This theorem is referenced by:  pr2dom  40235