Theorem pm110.643 8943

Description: 1+1=2 for cardinal number addition, derived from pm54.43 8770 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 8702), but after applying definitions, our theorem is equivalent. The comment for cdaval 8936 explains why we use ≈ instead of =. See pm110.643ALT 8944 for a shorter proof that doesn't use pm54.43 8770. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)

Assertion

Ref	Expression
pm110.643	⊢ (1𝑜 +𝑐 1𝑜) ≈ 2𝑜

Proof of Theorem pm110.643

Step	Hyp	Ref	Expression
1		1on 7512	. . 3 ⊢ 1𝑜 ∈ On
2		cdaval 8936	. . 3 ⊢ ((1𝑜 ∈ On ∧ 1𝑜 ∈ On) → (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})))
3	1, 1, 2	mp2an 707	. 2 ⊢ (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜}))
4		xp01disj 7521	. . 3 ⊢ ((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
5	1	elexi 3199	. . . . 5 ⊢ 1𝑜 ∈ V
6		0ex 4750	. . . . 5 ⊢ ∅ ∈ V
7	5, 6	xpsnen 7988	. . . 4 ⊢ (1𝑜 × {∅}) ≈ 1𝑜
8	5, 5	xpsnen 7988	. . . 4 ⊢ (1𝑜 × {1𝑜}) ≈ 1𝑜
9		pm54.43 8770	. . . 4 ⊢ (((1𝑜 × {∅}) ≈ 1𝑜 ∧ (1𝑜 × {1𝑜}) ≈ 1𝑜) → (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜))
10	7, 8, 9	mp2an 707	. . 3 ⊢ (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜)
11	4, 10	mpbi 220	. 2 ⊢ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜
12	3, 11	eqbrtri 4634	1 ⊢ (1𝑜 +𝑐 1𝑜) ≈ 2𝑜

Syntax hints:   ↔ wb 196   = wceq 1480   ∈ wcel 1987   ∪ cun 3553   ∩ cin 3554  ∅c0 3891  {csn 4148   class class class wbr 4613   × cxp 5072  Oncon0 5682  (class class class)co 6604  1_𝑜c1o 7498  2_𝑜c2o 7499   ≈ cen 7896   +_𝑐 ccda 8933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1o 7505  df-2o 7506  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-cda 8934

This theorem is referenced by: (None)