Theorem pm110.643 9200

Description: 1+1=2 for cardinal number addition, derived from pm54.43 9025 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 8921), but after applying definitions, our theorem is equivalent. The comment for cdaval 9193 explains why we use ≈ instead of =. See pm110.643ALT 9201 for a shorter proof that doesn't use pm54.43 9025. (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 7719	. . 3 ⊢ 1𝑜 ∈ On
2		cdaval 9193	. . 3 ⊢ ((1𝑜 ∈ On ∧ 1𝑜 ∈ On) → (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})))
3	1, 1, 2	mp2an 664	. 2 ⊢ (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜}))
4		xp01disj 7729	. . 3 ⊢ ((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
5	1	elexi 3364	. . . . 5 ⊢ 1𝑜 ∈ V
6		0ex 4924	. . . . 5 ⊢ ∅ ∈ V
7	5, 6	xpsnen 8199	. . . 4 ⊢ (1𝑜 × {∅}) ≈ 1𝑜
8	5, 5	xpsnen 8199	. . . 4 ⊢ (1𝑜 × {1𝑜}) ≈ 1𝑜
9		pm54.43 9025	. . . 4 ⊢ (((1𝑜 × {∅}) ≈ 1𝑜 ∧ (1𝑜 × {1𝑜}) ≈ 1𝑜) → (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜))
10	7, 8, 9	mp2an 664	. . 3 ⊢ (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜)
11	4, 10	mpbi 220	. 2 ⊢ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜
12	3, 11	eqbrtri 4807	1 ⊢ (1𝑜 +𝑐 1𝑜) ≈ 2𝑜

Syntax hints:   ↔ wb 196   = wceq 1631   ∈ wcel 2145   ∪ cun 3721   ∩ cin 3722  ∅c0 4063  {csn 4316   class class class wbr 4786   × cxp 5247  Oncon0 5866  (class class class)co 6792  1_𝑜c1o 7705  2_𝑜c2o 7706   ≈ cen 8105   +_𝑐 ccda 9190

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 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  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 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1o 7712  df-2o 7713  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-cda 9191

This theorem is referenced by: (None)