Theorem dju1p1e2 10085

Description: 1+1=2 for cardinal number addition, derived from pm54.43 9914 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 9808), but after applying definitions, our theorem is equivalent. Because we use a disjoint union for cardinal addition (as explained in the comment at the top of this section), we use ≈ instead of =. See dju1p1e2ALT 10086 for a shorter proof that doesn't use pm54.43 9914. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.)

Assertion

Ref	Expression
dju1p1e2	⊢ (1o ⊔ 1o) ≈ 2o

Proof of Theorem dju1p1e2

Step	Hyp	Ref	Expression
1		df-dju 9814	. 2 ⊢ (1o ⊔ 1o) = (({∅} × 1o) ∪ ({1o} × 1o))
2		xp01disjl 8416	. . 3 ⊢ (({∅} × 1o) ∩ ({1o} × 1o)) = ∅
3		0ex 5231	. . . . 5 ⊢ ∅ ∈ V
4		1on 8406	. . . . 5 ⊢ 1o ∈ On
5		xpsnen2g 8997	. . . . 5 ⊢ ((∅ ∈ V ∧ 1o ∈ On) → ({∅} × 1o) ≈ 1o)
6	3, 4, 5	mp2an 693	. . . 4 ⊢ ({∅} × 1o) ≈ 1o
7		xpsnen2g 8997	. . . . 5 ⊢ ((1o ∈ On ∧ 1o ∈ On) → ({1o} × 1o) ≈ 1o)
8	4, 4, 7	mp2an 693	. . . 4 ⊢ ({1o} × 1o) ≈ 1o
9		pm54.43 9914	. . . 4 ⊢ ((({∅} × 1o) ≈ 1o ∧ ({1o} × 1o) ≈ 1o) → ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o))
10	6, 8, 9	mp2an 693	. . 3 ⊢ ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o)
11	2, 10	mpbi 230	. 2 ⊢ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o
12	1, 11	eqbrtri 5095	1 ⊢ (1o ⊔ 1o) ≈ 2o

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  Vcvv 3427   ∪ cun 3883   ∩ cin 3884  ∅c0 4263  {csn 4557   class class class wbr 5074   × cxp 5618  Oncon0 6312  1_oc1o 8387  2_oc2o 8388   ≈ cen 8879   ⊔ cdju 9811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-ord 6315  df-on 6316  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-1st 7931  df-2nd 7932  df-1o 8394  df-2o 8395  df-er 8632  df-en 8883  df-dom 8884  df-sdom 8885  df-dju 9814

This theorem is referenced by:  pr2dom  43942