Theorem cda1en 3721

Description: Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143.

Hypothesis

Ref	Expression
cda0en.1	⊢ A ∈ V

Assertion

Ref	Expression
cda1en	⊢ (A +c 1o) ≈ suc (card ‘A)

Proof of Theorem cda1en

Step	Hyp	Ref	Expression
1		cda0en.1	. . . . 5 ⊢ A ∈ V
2		0ex 1745	. . . . . 6 ⊢ ∅ ∈ V
3	1, 2	xpsnen 3339	. . . . 5 ⊢ (A × {∅}) ≈ A
4		cardid 3635	. . . . 5 ⊢ (card ‘A) ≈ A
5	1, 3, 4	entr4 3324	. . . 4 ⊢ (A × {∅}) ≈ (card ‘A)
6		1o 3109	. . . . . 6 ⊢ 1o ∈ On
7	6	elisseti 1355	. . . . 5 ⊢ 1o ∈ V
8	7, 7	xpsnen 3339	. . . . 5 ⊢ (1o × {1o}) ≈ 1o
9		fvex 2838	. . . . . 6 ⊢ (card ‘A) ∈ V
10	9	ensn1 3329	. . . . 5 ⊢ {(card ‘A)} ≈ 1o
11	7, 8, 10	entr4 3324	. . . 4 ⊢ (1o × {1o}) ≈ {(card ‘A)}
12	5, 11	pm3.2i 234	. . 3 ⊢ ((A × {∅}) ≈ (card ‘A) ∧ (1o × {1o}) ≈ {(card ‘A)})
13		0ne1oOLD 3113	. . . . 5 ⊢ ¬ ∅ = 1o
14		xpsndisj 2655	. . . . 5 ⊢ (¬ ∅ = 1o → ((A × {∅}) ∩ (1o × {1o})) = ∅)
15	13, 14	ax-mp 6	. . . 4 ⊢ ((A × {∅}) ∩ (1o × {1o})) = ∅
16		cardon 3634	. . . . . 6 ⊢ (card ‘A) ∈ On
17	16	onord 2343	. . . . 5 ⊢ Ord (card ‘A)
18		orddisj 2236	. . . . 5 ⊢ (Ord (card ‘A) → ((card ‘A) ∩ {(card ‘A)}) = ∅)
19	17, 18	ax-mp 6	. . . 4 ⊢ ((card ‘A) ∩ {(card ‘A)}) = ∅
20	15, 19	pm3.2i 234	. . 3 ⊢ (((A × {∅}) ∩ (1o × {1o})) = ∅ ∧ ((card ‘A) ∩ {(card ‘A)}) = ∅)
21		unen 3338	. . 3 ⊢ ((((A × {∅}) ≈ (card ‘A) ∧ (1o × {1o}) ≈ {(card ‘A)}) ∧ (((A × {∅}) ∩ (1o × {1o})) = ∅ ∧ ((card ‘A) ∩ {(card ‘A)}) = ∅)) → ((A × {∅}) ∪ (1o × {1o})) ≈ ((card ‘A) ∪ {(card ‘A)}))
22	12, 20, 21	mp2an 520	. 2 ⊢ ((A × {∅}) ∪ (1o × {1o})) ≈ ((card ‘A) ∪ {(card ‘A)})
23	1, 7	cdaval 3717	. 2 ⊢ (A +c 1o) = ((A × {∅}) ∪ (1o × {1o}))
24		df-suc 2205	. 2 ⊢ suc (card ‘A) = ((card ‘A) ∪ {(card ‘A)})
25	22, 23, 24	3brtr4 2085	1 ⊢ (A +c 1o) ≈ suc (card ‘A)

Syntax hints:  ¬ wn 1   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485   ∩ cin 1486  ∅c0 1707  {csn 1808   class class class wbr 2054  Ord word 2198  Oncon0 2199  suc csuc 2201   × cxp 2408   ‘cfv 2422  (class class class)co 3001  1_oc1o 3099   ≈ cen 3271  cardccrd 3620   +_c ccda 3714

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  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  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-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  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-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  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-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-opr 3003  df-oprab 3004  df-1o 3104  df-er 3200  df-en 3274  df-card 3623  df-cda 3715

metamath.org