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Theorem cdaen 8955
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 ((𝐴𝐵𝐶𝐷) → (𝐴 +𝑐 𝐶) ≈ (𝐵 +𝑐 𝐷))

Proof of Theorem cdaen
StepHypRef Expression
1 relen 7920 . . . . . 6 Rel ≈
21brrelexi 5128 . . . . 5 (𝐴𝐵𝐴 ∈ V)
3 0ex 4760 . . . . 5 ∅ ∈ V
4 xpsneng 8005 . . . . 5 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
52, 3, 4sylancl 693 . . . 4 (𝐴𝐵 → (𝐴 × {∅}) ≈ 𝐴)
61brrelex2i 5129 . . . . . . 7 (𝐴𝐵𝐵 ∈ V)
7 xpsneng 8005 . . . . . . 7 ((𝐵 ∈ V ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
86, 3, 7sylancl 693 . . . . . 6 (𝐴𝐵 → (𝐵 × {∅}) ≈ 𝐵)
98ensymd 7967 . . . . 5 (𝐴𝐵𝐵 ≈ (𝐵 × {∅}))
10 entr 7968 . . . . 5 ((𝐴𝐵𝐵 ≈ (𝐵 × {∅})) → 𝐴 ≈ (𝐵 × {∅}))
119, 10mpdan 701 . . . 4 (𝐴𝐵𝐴 ≈ (𝐵 × {∅}))
12 entr 7968 . . . 4 (((𝐴 × {∅}) ≈ 𝐴𝐴 ≈ (𝐵 × {∅})) → (𝐴 × {∅}) ≈ (𝐵 × {∅}))
135, 11, 12syl2anc 692 . . 3 (𝐴𝐵 → (𝐴 × {∅}) ≈ (𝐵 × {∅}))
141brrelexi 5128 . . . . 5 (𝐶𝐷𝐶 ∈ V)
15 1on 7527 . . . . 5 1𝑜 ∈ On
16 xpsneng 8005 . . . . 5 ((𝐶 ∈ V ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
1714, 15, 16sylancl 693 . . . 4 (𝐶𝐷 → (𝐶 × {1𝑜}) ≈ 𝐶)
181brrelex2i 5129 . . . . . . 7 (𝐶𝐷𝐷 ∈ V)
19 xpsneng 8005 . . . . . . 7 ((𝐷 ∈ V ∧ 1𝑜 ∈ On) → (𝐷 × {1𝑜}) ≈ 𝐷)
2018, 15, 19sylancl 693 . . . . . 6 (𝐶𝐷 → (𝐷 × {1𝑜}) ≈ 𝐷)
2120ensymd 7967 . . . . 5 (𝐶𝐷𝐷 ≈ (𝐷 × {1𝑜}))
22 entr 7968 . . . . 5 ((𝐶𝐷𝐷 ≈ (𝐷 × {1𝑜})) → 𝐶 ≈ (𝐷 × {1𝑜}))
2321, 22mpdan 701 . . . 4 (𝐶𝐷𝐶 ≈ (𝐷 × {1𝑜}))
24 entr 7968 . . . 4 (((𝐶 × {1𝑜}) ≈ 𝐶𝐶 ≈ (𝐷 × {1𝑜})) → (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜}))
2517, 23, 24syl2anc 692 . . 3 (𝐶𝐷 → (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜}))
26 xp01disj 7536 . . . 4 ((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
27 xp01disj 7536 . . . 4 ((𝐵 × {∅}) ∩ (𝐷 × {1𝑜})) = ∅
28 unen 8000 . . . 4 ((((𝐴 × {∅}) ≈ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜})) ∧ (((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅ ∧ ((𝐵 × {∅}) ∩ (𝐷 × {1𝑜})) = ∅)) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≈ ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
2926, 27, 28mpanr12 720 . . 3 (((𝐴 × {∅}) ≈ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜})) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≈ ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
3013, 25, 29syl2an 494 . 2 ((𝐴𝐵𝐶𝐷) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≈ ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
31 cdaval 8952 . . 3 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
322, 14, 31syl2an 494 . 2 ((𝐴𝐵𝐶𝐷) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
33 cdaval 8952 . . 3 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 +𝑐 𝐷) = ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
346, 18, 33syl2an 494 . 2 ((𝐴𝐵𝐶𝐷) → (𝐵 +𝑐 𝐷) = ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
3530, 32, 343brtr4d 4655 1 ((𝐴𝐵𝐶𝐷) → (𝐴 +𝑐 𝐶) ≈ (𝐵 +𝑐 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  cun 3558  cin 3559  c0 3897  {csn 4155   class class class wbr 4623   × cxp 5082  Oncon0 5692  (class class class)co 6615  1𝑜c1o 7513  cen 7912   +𝑐 ccda 8949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-ord 5695  df-on 5696  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1o 7520  df-er 7702  df-en 7916  df-cda 8950
This theorem is referenced by:  cdaenun  8956  cardacda  8980  pwsdompw  8986  ackbij1lem5  9006  ackbij1lem9  9010  gchhar  9461
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