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Theorem cdaen 9196
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 8113 . . . . . 6 Rel ≈
21brrelexi 5298 . . . . 5 (𝐴𝐵𝐴 ∈ V)
3 0ex 4921 . . . . 5 ∅ ∈ V
4 xpsneng 8200 . . . . 5 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
52, 3, 4sylancl 566 . . . 4 (𝐴𝐵 → (𝐴 × {∅}) ≈ 𝐴)
61brrelex2i 5299 . . . . . . 7 (𝐴𝐵𝐵 ∈ V)
7 xpsneng 8200 . . . . . . 7 ((𝐵 ∈ V ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
86, 3, 7sylancl 566 . . . . . 6 (𝐴𝐵 → (𝐵 × {∅}) ≈ 𝐵)
98ensymd 8159 . . . . 5 (𝐴𝐵𝐵 ≈ (𝐵 × {∅}))
10 entr 8160 . . . . 5 ((𝐴𝐵𝐵 ≈ (𝐵 × {∅})) → 𝐴 ≈ (𝐵 × {∅}))
119, 10mpdan 659 . . . 4 (𝐴𝐵𝐴 ≈ (𝐵 × {∅}))
12 entr 8160 . . . 4 (((𝐴 × {∅}) ≈ 𝐴𝐴 ≈ (𝐵 × {∅})) → (𝐴 × {∅}) ≈ (𝐵 × {∅}))
135, 11, 12syl2anc 565 . . 3 (𝐴𝐵 → (𝐴 × {∅}) ≈ (𝐵 × {∅}))
141brrelexi 5298 . . . . 5 (𝐶𝐷𝐶 ∈ V)
15 1on 7719 . . . . 5 1𝑜 ∈ On
16 xpsneng 8200 . . . . 5 ((𝐶 ∈ V ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
1714, 15, 16sylancl 566 . . . 4 (𝐶𝐷 → (𝐶 × {1𝑜}) ≈ 𝐶)
181brrelex2i 5299 . . . . . . 7 (𝐶𝐷𝐷 ∈ V)
19 xpsneng 8200 . . . . . . 7 ((𝐷 ∈ V ∧ 1𝑜 ∈ On) → (𝐷 × {1𝑜}) ≈ 𝐷)
2018, 15, 19sylancl 566 . . . . . 6 (𝐶𝐷 → (𝐷 × {1𝑜}) ≈ 𝐷)
2120ensymd 8159 . . . . 5 (𝐶𝐷𝐷 ≈ (𝐷 × {1𝑜}))
22 entr 8160 . . . . 5 ((𝐶𝐷𝐷 ≈ (𝐷 × {1𝑜})) → 𝐶 ≈ (𝐷 × {1𝑜}))
2321, 22mpdan 659 . . . 4 (𝐶𝐷𝐶 ≈ (𝐷 × {1𝑜}))
24 entr 8160 . . . 4 (((𝐶 × {1𝑜}) ≈ 𝐶𝐶 ≈ (𝐷 × {1𝑜})) → (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜}))
2517, 23, 24syl2anc 565 . . 3 (𝐶𝐷 → (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜}))
26 xp01disj 7729 . . . 4 ((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
27 xp01disj 7729 . . . 4 ((𝐵 × {∅}) ∩ (𝐷 × {1𝑜})) = ∅
28 unen 8195 . . . 4 ((((𝐴 × {∅}) ≈ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜})) ∧ (((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅ ∧ ((𝐵 × {∅}) ∩ (𝐷 × {1𝑜})) = ∅)) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≈ ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
2926, 27, 28mpanr12 677 . . 3 (((𝐴 × {∅}) ≈ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜})) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≈ ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
3013, 25, 29syl2an 575 . 2 ((𝐴𝐵𝐶𝐷) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≈ ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
31 cdaval 9193 . . 3 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
322, 14, 31syl2an 575 . 2 ((𝐴𝐵𝐶𝐷) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
33 cdaval 9193 . . 3 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 +𝑐 𝐷) = ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
346, 18, 33syl2an 575 . 2 ((𝐴𝐵𝐶𝐷) → (𝐵 +𝑐 𝐷) = ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
3530, 32, 343brtr4d 4816 1 ((𝐴𝐵𝐶𝐷) → (𝐴 +𝑐 𝐶) ≈ (𝐵 +𝑐 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  cun 3719  cin 3720  c0 4061  {csn 4314   class class class wbr 4784   × cxp 5247  Oncon0 5866  (class class class)co 6792  1𝑜c1o 7705  cen 8105   +𝑐 ccda 9190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1o 7712  df-er 7895  df-en 8109  df-cda 9191
This theorem is referenced by:  cdaenun  9197  cardacda  9221  pwsdompw  9227  ackbij1lem5  9247  ackbij1lem9  9251  gchhar  9702
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