Theorem cdaval 3717

Description: Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258. For cardinal arithmetic, we follow Mendelson. Rather than defining operations restricted to cardinal numbers, we use this disjoint union operation for addition, while cross product and set exponentiation stand in for cardinal multiplication and exponentiation. Equinumerosity and dominance serve the roles of equality and ordering. If we wanted to, we could easily convert our theorems to actual cardinal number operations via carden 3638, carddom 3642, and cardsdom 3643. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available.

Hypotheses

Ref	Expression
cdaval.1	⊢ A ∈ V
cdaval.2	⊢ B ∈ V

Assertion

Ref	Expression
cdaval	⊢ (A +c B) = ((A × {∅}) ∪ (B × {1o}))

Proof of Theorem cdaval

Step	Hyp	Ref	Expression
1		cdaval.1	. 2 ⊢ A ∈ V
2		cdaval.2	. 2 ⊢ B ∈ V
3		cdavalt 3716	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A +c B) = ((A × {∅}) ∪ (B × {1o})))
4	1, 2, 3	mp2an 520	1 ⊢ (A +c B) = ((A × {∅}) ∪ (B × {1o}))

Syntax hints:   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485  ∅c0 1707  {csn 1808   × cxp 2408  (class class class)co 3001  1_oc1o 3099   +_c ccda 3714

This theorem is referenced by:  uncdadom 3718  cdaen 3719  cda0en 3720  cda1en 3721  xp2cda 3723  cdacomen 3724  cdaassen 3725  xpcdaen 3726  cdadom1 3727

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003  df-oprab 3004  df-cda 3715

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