Theorem cdaen 8995

Description: Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
cdaen	⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 +𝑐 𝐶) ≈ (𝐵 +𝑐 𝐷))

Proof of Theorem cdaen

Step	Hyp	Ref	Expression
1		relen 7960	. . . . . 6 ⊢ Rel ≈
2	1	brrelexi 5158	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V)
3		0ex 4790	. . . . 5 ⊢ ∅ ∈ V
4		xpsneng 8045	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
5	2, 3, 4	sylancl 694	. . . 4 ⊢ (𝐴 ≈ 𝐵 → (𝐴 × {∅}) ≈ 𝐴)
6	1	brrelex2i 5159	. . . . . . 7 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V)
7		xpsneng 8045	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
8	6, 3, 7	sylancl 694	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (𝐵 × {∅}) ≈ 𝐵)
9	8	ensymd 8007	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ (𝐵 × {∅}))
10		entr 8008	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ (𝐵 × {∅})) → 𝐴 ≈ (𝐵 × {∅}))
11	9, 10	mpdan 702	. . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≈ (𝐵 × {∅}))
12		entr 8008	. . . 4 ⊢ (((𝐴 × {∅}) ≈ 𝐴 ∧ 𝐴 ≈ (𝐵 × {∅})) → (𝐴 × {∅}) ≈ (𝐵 × {∅}))
13	5, 11, 12	syl2anc 693	. . 3 ⊢ (𝐴 ≈ 𝐵 → (𝐴 × {∅}) ≈ (𝐵 × {∅}))
14	1	brrelexi 5158	. . . . 5 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ∈ V)
15		1on 7567	. . . . 5 ⊢ 1𝑜 ∈ On
16		xpsneng 8045	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
17	14, 15, 16	sylancl 694	. . . 4 ⊢ (𝐶 ≈ 𝐷 → (𝐶 × {1𝑜}) ≈ 𝐶)
18	1	brrelex2i 5159	. . . . . . 7 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ∈ V)
19		xpsneng 8045	. . . . . . 7 ⊢ ((𝐷 ∈ V ∧ 1𝑜 ∈ On) → (𝐷 × {1𝑜}) ≈ 𝐷)
20	18, 15, 19	sylancl 694	. . . . . 6 ⊢ (𝐶 ≈ 𝐷 → (𝐷 × {1𝑜}) ≈ 𝐷)
21	20	ensymd 8007	. . . . 5 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ≈ (𝐷 × {1𝑜}))
22		entr 8008	. . . . 5 ⊢ ((𝐶 ≈ 𝐷 ∧ 𝐷 ≈ (𝐷 × {1𝑜})) → 𝐶 ≈ (𝐷 × {1𝑜}))
23	21, 22	mpdan 702	. . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ≈ (𝐷 × {1𝑜}))
24		entr 8008	. . . 4 ⊢ (((𝐶 × {1𝑜}) ≈ 𝐶 ∧ 𝐶 ≈ (𝐷 × {1𝑜})) → (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜}))
25	17, 23, 24	syl2anc 693	. . 3 ⊢ (𝐶 ≈ 𝐷 → (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜}))
26		xp01disj 7576	. . . 4 ⊢ ((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
27		xp01disj 7576	. . . 4 ⊢ ((𝐵 × {∅}) ∩ (𝐷 × {1𝑜})) = ∅
28		unen 8040	. . . 4 ⊢ ((((𝐴 × {∅}) ≈ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜})) ∧ (((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅ ∧ ((𝐵 × {∅}) ∩ (𝐷 × {1𝑜})) = ∅)) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≈ ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
29	26, 27, 28	mpanr12 721	. . 3 ⊢ (((𝐴 × {∅}) ≈ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜})) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≈ ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
30	13, 25, 29	syl2an 494	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≈ ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
31		cdaval 8992	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
32	2, 14, 31	syl2an 494	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
33		cdaval 8992	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 +𝑐 𝐷) = ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
34	6, 18, 33	syl2an 494	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 +𝑐 𝐷) = ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
35	30, 32, 34	3brtr4d 4685	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 +𝑐 𝐶) ≈ (𝐵 +𝑐 𝐷))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572   ∩ cin 3573  ∅c0 3915  {csn 4177   class class class wbr 4653   × cxp 5112  Oncon0 5723  (class class class)co 6650  1_𝑜c1o 7553   ≈ cen 7952   +_𝑐 ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-er 7742  df-en 7956  df-cda 8990

This theorem is referenced by:  cdaenun  8996  cardacda  9020  pwsdompw  9026  ackbij1lem5  9046  ackbij1lem9  9050  gchhar  9501