Theorem pm110.643 8692

Description: 1+1=2 for cardinal number addition, derived from pm54.43 8519 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 8451), but after applying definitions, our theorem is equivalent. The comment for cdaval 8685 explains why we use ≈ instead of =. See pm110.643ALT 8693 for a shorter proof that doesn't use pm54.43 8519. (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 7266	. . 3 ⊢ 1𝑜 ∈ On
2		cdaval 8685	. . 3 ⊢ ((1𝑜 ∈ On ∧ 1𝑜 ∈ On) → (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})))
3	1, 1, 2	mp2an 695	. 2 ⊢ (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜}))
4		xp01disj 7275	. . 3 ⊢ ((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
5	1	elexi 3076	. . . . 5 ⊢ 1𝑜 ∈ V
6		0ex 4568	. . . . 5 ⊢ ∅ ∈ V
7	5, 6	xpsnen 7740	. . . 4 ⊢ (1𝑜 × {∅}) ≈ 1𝑜
8	5, 5	xpsnen 7740	. . . 4 ⊢ (1𝑜 × {1𝑜}) ≈ 1𝑜
9		pm54.43 8519	. . . 4 ⊢ (((1𝑜 × {∅}) ≈ 1𝑜 ∧ (1𝑜 × {1𝑜}) ≈ 1𝑜) → (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜))
10	7, 8, 9	mp2an 695	. . 3 ⊢ (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜)
11	4, 10	mpbi 215	. 2 ⊢ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜
12	3, 11	eqbrtri 4454	1 ⊢ (1𝑜 +𝑐 1𝑜) ≈ 2𝑜

Syntax hints:   ↔ wb 191   = wceq 1468   ∈ wcel 1937   ∪ cun 3424   ∩ cin 3425  ∅c0 3757  {csn 3995   class class class wbr 4434   × cxp 4878  Oncon0 5474  (class class class)co 6363  1_𝑜c1o 7252  2_𝑜c2o 7253   ≈ cen 7649   +_𝑐 ccda 8682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680  ax-un 6659

This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3068  df-sbc 3292  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-int 4265  df-br 4435  df-opab 4494  df-mpt 4495  df-tr 4531  df-eprel 4791  df-id 4795  df-po 4801  df-so 4802  df-fr 4839  df-we 4841  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-ord 5477  df-on 5478  df-lim 5479  df-suc 5480  df-iota 5597  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-fv 5641  df-ov 6366  df-oprab 6367  df-mpt2 6368  df-om 6770  df-1o 7259  df-2o 7260  df-er 7440  df-en 7653  df-dom 7654  df-sdom 7655  df-cda 8683

This theorem is referenced by: (None)