Theorem pm110.643 9183

Description: 1+1=2 for cardinal number addition, derived from pm54.43 9008 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 8923), but after applying definitions, our theorem is equivalent. The comment for cdaval 9176 explains why we use ≈ instead of =. See pm110.643ALT 9184 for a shorter proof that doesn't use pm54.43 9008. (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 7728	. . 3 ⊢ 1𝑜 ∈ On
2		cdaval 9176	. . 3 ⊢ ((1𝑜 ∈ On ∧ 1𝑜 ∈ On) → (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})))
3	1, 1, 2	mp2an 710	. 2 ⊢ (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜}))
4		xp01disj 7737	. . 3 ⊢ ((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
5	1	elexi 3345	. . . . 5 ⊢ 1𝑜 ∈ V
6		0ex 4934	. . . . 5 ⊢ ∅ ∈ V
7	5, 6	xpsnen 8201	. . . 4 ⊢ (1𝑜 × {∅}) ≈ 1𝑜
8	5, 5	xpsnen 8201	. . . 4 ⊢ (1𝑜 × {1𝑜}) ≈ 1𝑜
9		pm54.43 9008	. . . 4 ⊢ (((1𝑜 × {∅}) ≈ 1𝑜 ∧ (1𝑜 × {1𝑜}) ≈ 1𝑜) → (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜))
10	7, 8, 9	mp2an 710	. . 3 ⊢ (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜)
11	4, 10	mpbi 220	. 2 ⊢ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜
12	3, 11	eqbrtri 4817	1 ⊢ (1𝑜 +𝑐 1𝑜) ≈ 2𝑜

Syntax hints:   ↔ wb 196   = wceq 1624   ∈ wcel 2131   ∪ cun 3705   ∩ cin 3706  ∅c0 4050  {csn 4313   class class class wbr 4796   × cxp 5256  Oncon0 5876  (class class class)co 6805  1_𝑜c1o 7714  2_𝑜c2o 7715   ≈ cen 8110   +_𝑐 ccda 9173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1o 7721  df-2o 7722  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-cda 9174

This theorem is referenced by: (None)