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Theorem cda1dif 9036
 Description: Adding and subtracting one gives back the original set. Similar to pncan 10325 for cardinalities. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
cda1dif (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)

Proof of Theorem cda1dif
StepHypRef Expression
1 ovexd 6720 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) ∈ V)
2 id 22 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → 𝐵 ∈ (𝐴 +𝑐 1𝑜))
3 df1o2 7617 . . . . . . . 8 1𝑜 = {∅}
43xpeq1i 5169 . . . . . . 7 (1𝑜 × {1𝑜}) = ({∅} × {1𝑜})
5 0ex 4823 . . . . . . . 8 ∅ ∈ V
6 1on 7612 . . . . . . . . 9 1𝑜 ∈ On
76elexi 3244 . . . . . . . 8 1𝑜 ∈ V
85, 7xpsn 6447 . . . . . . 7 ({∅} × {1𝑜}) = {⟨∅, 1𝑜⟩}
94, 8eqtri 2673 . . . . . 6 (1𝑜 × {1𝑜}) = {⟨∅, 1𝑜⟩}
10 ssun2 3810 . . . . . 6 (1𝑜 × {1𝑜}) ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
119, 10eqsstr3i 3669 . . . . 5 {⟨∅, 1𝑜⟩} ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
12 opex 4962 . . . . . 6 ⟨∅, 1𝑜⟩ ∈ V
1312snss 4348 . . . . 5 (⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ↔ {⟨∅, 1𝑜⟩} ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
1411, 13mpbir 221 . . . 4 ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
15 relxp 5160 . . . . . . . 8 Rel (V × V)
16 cdafn 9029 . . . . . . . . . 10 +𝑐 Fn (V × V)
17 fndm 6028 . . . . . . . . . 10 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
1816, 17ax-mp 5 . . . . . . . . 9 dom +𝑐 = (V × V)
1918releqi 5236 . . . . . . . 8 (Rel dom +𝑐 ↔ Rel (V × V))
2015, 19mpbir 221 . . . . . . 7 Rel dom +𝑐
2120ovrcl 6726 . . . . . 6 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ V ∧ 1𝑜 ∈ V))
2221simpld 474 . . . . 5 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
23 cdaval 9030 . . . . 5 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
2422, 6, 23sylancl 695 . . . 4 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
2514, 24syl5eleqr 2737 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜))
26 difsnen 8083 . . 3 (((𝐴 +𝑐 1𝑜) ∈ V ∧ 𝐵 ∈ (𝐴 +𝑐 1𝑜) ∧ ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜)) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}))
271, 2, 25, 26syl3anc 1366 . 2 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}))
2824difeq1d 3760 . . . 4 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩}))
29 xp01disj 7621 . . . . . 6 ((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
30 disj3 4054 . . . . . 6 (((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜})))
3129, 30mpbi 220 . . . . 5 (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
32 difun2 4081 . . . . 5 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
339difeq2i 3758 . . . . 5 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩})
3431, 32, 333eqtr2i 2679 . . . 4 (𝐴 × {∅}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩})
3528, 34syl6eqr 2703 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (𝐴 × {∅}))
36 xpsneng 8086 . . . 4 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
3722, 5, 36sylancl 695 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ 𝐴)
3835, 37eqbrtrd 4707 . 2 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ 𝐴)
39 entr 8049 . 2 ((((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ∧ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ 𝐴) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)
4027, 38, 39syl2anc 694 1 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∖ cdif 3604   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  {csn 4210  ⟨cop 4216   class class class wbr 4685   × cxp 5141  dom cdm 5143  Rel wrel 5148  Oncon0 5761   Fn wfn 5921  (class class class)co 6690  1𝑜c1o 7598   ≈ cen 7994   +𝑐 ccda 9027 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-1o 7605  df-er 7787  df-en 7998  df-cda 9028 This theorem is referenced by:  canthp1  9514
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