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Theorem cdaen 9284
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 8201 . . . . . 6 Rel ≈
21brrelex1i 5364 . . . . 5 (𝐴𝐵𝐴 ∈ V)
3 0ex 4985 . . . . 5 ∅ ∈ V
4 xpsneng 8288 . . . . 5 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
52, 3, 4sylancl 581 . . . 4 (𝐴𝐵 → (𝐴 × {∅}) ≈ 𝐴)
61brrelex2i 5365 . . . . . . 7 (𝐴𝐵𝐵 ∈ V)
7 xpsneng 8288 . . . . . . 7 ((𝐵 ∈ V ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
86, 3, 7sylancl 581 . . . . . 6 (𝐴𝐵 → (𝐵 × {∅}) ≈ 𝐵)
98ensymd 8247 . . . . 5 (𝐴𝐵𝐵 ≈ (𝐵 × {∅}))
10 entr 8248 . . . . 5 ((𝐴𝐵𝐵 ≈ (𝐵 × {∅})) → 𝐴 ≈ (𝐵 × {∅}))
119, 10mpdan 679 . . . 4 (𝐴𝐵𝐴 ≈ (𝐵 × {∅}))
12 entr 8248 . . . 4 (((𝐴 × {∅}) ≈ 𝐴𝐴 ≈ (𝐵 × {∅})) → (𝐴 × {∅}) ≈ (𝐵 × {∅}))
135, 11, 12syl2anc 580 . . 3 (𝐴𝐵 → (𝐴 × {∅}) ≈ (𝐵 × {∅}))
141brrelex1i 5364 . . . . 5 (𝐶𝐷𝐶 ∈ V)
15 1on 7807 . . . . 5 1𝑜 ∈ On
16 xpsneng 8288 . . . . 5 ((𝐶 ∈ V ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
1714, 15, 16sylancl 581 . . . 4 (𝐶𝐷 → (𝐶 × {1𝑜}) ≈ 𝐶)
181brrelex2i 5365 . . . . . . 7 (𝐶𝐷𝐷 ∈ V)
19 xpsneng 8288 . . . . . . 7 ((𝐷 ∈ V ∧ 1𝑜 ∈ On) → (𝐷 × {1𝑜}) ≈ 𝐷)
2018, 15, 19sylancl 581 . . . . . 6 (𝐶𝐷 → (𝐷 × {1𝑜}) ≈ 𝐷)
2120ensymd 8247 . . . . 5 (𝐶𝐷𝐷 ≈ (𝐷 × {1𝑜}))
22 entr 8248 . . . . 5 ((𝐶𝐷𝐷 ≈ (𝐷 × {1𝑜})) → 𝐶 ≈ (𝐷 × {1𝑜}))
2321, 22mpdan 679 . . . 4 (𝐶𝐷𝐶 ≈ (𝐷 × {1𝑜}))
24 entr 8248 . . . 4 (((𝐶 × {1𝑜}) ≈ 𝐶𝐶 ≈ (𝐷 × {1𝑜})) → (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜}))
2517, 23, 24syl2anc 580 . . 3 (𝐶𝐷 → (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜}))
26 xp01disj 7817 . . . 4 ((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
27 xp01disj 7817 . . . 4 ((𝐵 × {∅}) ∩ (𝐷 × {1𝑜})) = ∅
28 unen 8283 . . . 4 ((((𝐴 × {∅}) ≈ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜})) ∧ (((𝐴 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅ ∧ ((𝐵 × {∅}) ∩ (𝐷 × {1𝑜})) = ∅)) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≈ ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
2926, 27, 28mpanr12 697 . . 3 (((𝐴 × {∅}) ≈ (𝐵 × {∅}) ∧ (𝐶 × {1𝑜}) ≈ (𝐷 × {1𝑜})) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≈ ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
3013, 25, 29syl2an 590 . 2 ((𝐴𝐵𝐶𝐷) → ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})) ≈ ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
31 cdaval 9281 . . 3 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
322, 14, 31syl2an 590 . 2 ((𝐴𝐵𝐶𝐷) → (𝐴 +𝑐 𝐶) = ((𝐴 × {∅}) ∪ (𝐶 × {1𝑜})))
33 cdaval 9281 . . 3 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 +𝑐 𝐷) = ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
346, 18, 33syl2an 590 . 2 ((𝐴𝐵𝐶𝐷) → (𝐵 +𝑐 𝐷) = ((𝐵 × {∅}) ∪ (𝐷 × {1𝑜})))
3530, 32, 343brtr4d 4876 1 ((𝐴𝐵𝐶𝐷) → (𝐴 +𝑐 𝐶) ≈ (𝐵 +𝑐 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3386  cun 3768  cin 3769  c0 4116  {csn 4369   class class class wbr 4844   × cxp 5311  Oncon0 5942  (class class class)co 6879  1𝑜c1o 7793  cen 8193   +𝑐 ccda 9278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-int 4669  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5221  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-we 5274  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-ord 5945  df-on 5946  df-suc 5948  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-1o 7800  df-er 7983  df-en 8197  df-cda 9279
This theorem is referenced by:  cdaenun  9285  cardacda  9309  pwsdompw  9315  ackbij1lem5  9335  ackbij1lem9  9339  gchhar  9790
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