Theorem pm110.643 9202

Description: 1+1=2 for cardinal number addition, derived from pm54.43 9027 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 9195 explains why we use ≈ instead of =. See pm110.643ALT 9203 for a shorter proof that doesn't use pm54.43 9027. (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 7721	. . 3 ⊢ 1𝑜 ∈ On
2		cdaval 9195	. . 3 ⊢ ((1𝑜 ∈ On ∧ 1𝑜 ∈ On) → (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})))
3	1, 1, 2	mp2an 666	. 2 ⊢ (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜}))
4		xp01disj 7731	. . 3 ⊢ ((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
5	1	elexi 3365	. . . . 5 ⊢ 1𝑜 ∈ V
6		0ex 4925	. . . . 5 ⊢ ∅ ∈ V
7	5, 6	xpsnen 8201	. . . 4 ⊢ (1𝑜 × {∅}) ≈ 1𝑜
8	5, 5	xpsnen 8201	. . . 4 ⊢ (1𝑜 × {1𝑜}) ≈ 1𝑜
9		pm54.43 9027	. . . 4 ⊢ (((1𝑜 × {∅}) ≈ 1𝑜 ∧ (1𝑜 × {1𝑜}) ≈ 1𝑜) → (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜))
10	7, 8, 9	mp2an 666	. . 3 ⊢ (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜)
11	4, 10	mpbi 220	. 2 ⊢ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜
12	3, 11	eqbrtri 4808	1 ⊢ (1𝑜 +𝑐 1𝑜) ≈ 2𝑜

Syntax hints:   ↔ wb 196   = wceq 1631   ∈ wcel 2145   ∪ cun 3722   ∩ cin 3723  ∅c0 4064  {csn 4317   class class class wbr 4787   × cxp 5248  Oncon0 5867  (class class class)co 6794  1_𝑜c1o 7707  2_𝑜c2o 7708   ≈ cen 8107   +_𝑐 ccda 9192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-1o 7714  df-2o 7715  df-er 7897  df-en 8111  df-dom 8112  df-sdom 8113  df-cda 9193

This theorem is referenced by: (None)