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Theorem cdaen 8878
 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 7846 . . . . . 6 Rel ≈
21brrelexi 5082 . . . . 5 (𝐴𝐵𝐴 ∈ V)
3 0ex 4718 . . . . 5 ∅ ∈ V
4 xpsneng 7930 . . . . 5 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
52, 3, 4sylancl 693 . . . 4 (𝐴𝐵 → (𝐴 × {∅}) ≈ 𝐴)
61brrelex2i 5083 . . . . . . 7 (𝐴𝐵𝐵 ∈ V)
7 xpsneng 7930 . . . . . . 7 ((𝐵 ∈ V ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
86, 3, 7sylancl 693 . . . . . 6 (𝐴𝐵 → (𝐵 × {∅}) ≈ 𝐵)
98ensymd 7893 . . . . 5 (𝐴𝐵𝐵 ≈ (𝐵 × {∅}))
10 entr 7894 . . . . 5 ((𝐴𝐵𝐵 ≈ (𝐵 × {∅})) → 𝐴 ≈ (𝐵 × {∅}))
119, 10mpdan 699 . . . 4 (𝐴𝐵𝐴 ≈ (𝐵 × {∅}))
12 entr 7894 . . . 4 (((𝐴 × {∅}) ≈ 𝐴𝐴 ≈ (𝐵 × {∅})) → (𝐴 × {∅}) ≈ (𝐵 × {∅}))
135, 11, 12syl2anc 691 . . 3 (𝐴𝐵 → (𝐴 × {∅}) ≈ (𝐵 × {∅}))
141brrelexi 5082 . . . . 5 (𝐶𝐷𝐶 ∈ V)
15 1on 7454 . . . . 5 1𝑜 ∈ On
16 xpsneng 7930 . . . . 5 ((𝐶 ∈ V ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
1714, 15, 16sylancl 693 . . . 4 (𝐶𝐷 → (𝐶 × {1𝑜}) ≈ 𝐶)
181brrelex2i 5083 . . . . . . 7 (𝐶𝐷𝐷 ∈ V)
19 xpsneng 7930 . . . . . . 7 ((𝐷 ∈ V ∧ 1𝑜 ∈ On) → (𝐷 × {1𝑜}) ≈ 𝐷)
2018, 15, 19sylancl 693 . . . . . 6 (𝐶𝐷 → (𝐷 × {1𝑜}) ≈ 𝐷)
2120ensymd 7893 . . . . 5 (𝐶𝐷𝐷 ≈ (𝐷 × {1𝑜}))
22 entr 7894 . . . . 5 ((𝐶𝐷𝐷 ≈ (𝐷 × {1𝑜})) → 𝐶 ≈ (𝐷 × {1𝑜}))
2321, 22mpdan 699 . . . 4 (𝐶𝐷𝐶 ≈ (𝐷 × {1𝑜}))
24 entr 7894 . . . 4 (((𝐶 × {1𝑜}) ≈ 𝐶𝐶 ≈ (𝐷 × {1𝑜})) → (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜}))
2517, 23, 24syl2anc 691 . . 3 (𝐶𝐷 → (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜}))
26 xp01disj 7463 . . . 4 ((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
27 xp01disj 7463 . . . 4 ((𝐵 × {∅}) ∩ (𝐷 × {1𝑜})) = ∅
28 unen 7925 . . . 4 ((((𝐴 × {∅}) ≈ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜})) ∧ (((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅ ∧ ((𝐵 × {∅}) ∩ (𝐷 × {1𝑜})) = ∅)) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≈ ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
2926, 27, 28mpanr12 717 . . 3 (((𝐴 × {∅}) ≈ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜})) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≈ ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
3013, 25, 29syl2an 493 . 2 ((𝐴𝐵𝐶𝐷) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≈ ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
31 cdaval 8875 . . 3 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
322, 14, 31syl2an 493 . 2 ((𝐴𝐵𝐶𝐷) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
33 cdaval 8875 . . 3 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 +𝑐 𝐷) = ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
346, 18, 33syl2an 493 . 2 ((𝐴𝐵𝐶𝐷) → (𝐵 +𝑐 𝐷) = ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
3530, 32, 343brtr4d 4615 1 ((𝐴𝐵𝐶𝐷) → (𝐴 +𝑐 𝐶) ≈ (𝐵 +𝑐 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  {csn 4125   class class class wbr 4583   × cxp 5036  Oncon0 5640  (class class class)co 6549  1𝑜c1o 7440   ≈ cen 7838   +𝑐 ccda 8872 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1o 7447  df-er 7629  df-en 7842  df-cda 8873 This theorem is referenced by:  cdaenun  8879  cardacda  8903  pwsdompw  8909  ackbij1lem5  8929  ackbij1lem9  8933  gchhar  9380
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