Theorem xp2cda 4940

Description: Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258.

Hypothesis

Ref	Expression
cda0en.1	⊢ A ∈ V

Assertion

Ref	Expression
xp2cda	⊢ (A × 2o) = (A +c A)

Proof of Theorem xp2cda

Step	Hyp	Ref	Expression
1		xpundi 3231	. 2 ⊢ (A × ({∅} ∪ {1o})) = ((A × {∅}) ∪ (A × {1o}))
2		df-pr 2417	. . . 4 ⊢ {∅, {∅}} = ({∅} ∪ {{∅}})
3		df2o2 4147	. . . 4 ⊢ 2o = {∅, {∅}}
4		df1o2 4146	. . . . . 6 ⊢ 1o = {∅}
5	4	sneqi 2422	. . . . 5 ⊢ {1o} = {{∅}}
6	5	uneq2i 2184	. . . 4 ⊢ ({∅} ∪ {1o}) = ({∅} ∪ {{∅}})
7	2, 3, 6	3eqtr4 1508	. . 3 ⊢ 2o = ({∅} ∪ {1o})
8		xpeq2 3207	. . 3 ⊢ (2o = ({∅} ∪ {1o}) → (A × 2o) = (A × ({∅} ∪ {1o})))
9	7, 8	ax-mp 7	. 2 ⊢ (A × 2o) = (A × ({∅} ∪ {1o}))
10		cda0en.1	. . 3 ⊢ A ∈ V
11	10, 10	cdaval 4932	. 2 ⊢ (A +c A) = ((A × {∅}) ∪ (A × {1o}))
12	1, 9, 11	3eqtr4 1508	1 ⊢ (A × 2o) = (A +c A)

Syntax hints:   = wceq 958   ∈ wcel 960  Vcvv 1814   ∪ cun 2048  ∅c0 2283  {csn 2413  {cpr 2414   × cxp 3174  (class class class)co 3969  1_oc1o 4134  2_oc2o 4135   +_c ccda 4929

This theorem is referenced by:  infunabs 7566  infcdaabs 7567

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-suc 2960  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-opr 3971  df-oprab 3972  df-1o 4139  df-2o 4140  df-cda 4930

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