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Theorem cda1dif 9221
Description: Adding and subtracting one gives back the original set. Similar to pncan 10510 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 6846 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) ∈ V)
2 id 22 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → 𝐵 ∈ (𝐴 +𝑐 1𝑜))
3 df1o2 7747 . . . . . . . 8 1𝑜 = {∅}
43xpeq1i 5288 . . . . . . 7 (1𝑜 × {1𝑜}) = ({∅} × {1𝑜})
5 0ex 4937 . . . . . . . 8 ∅ ∈ V
6 1oex 7742 . . . . . . . 8 1𝑜 ∈ V
75, 6xpsn 6568 . . . . . . 7 ({∅} × {1𝑜}) = {⟨∅, 1𝑜⟩}
84, 7eqtri 2796 . . . . . 6 (1𝑜 × {1𝑜}) = {⟨∅, 1𝑜⟩}
9 ssun2 3935 . . . . . 6 (1𝑜 × {1𝑜}) ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
108, 9eqsstr3i 3792 . . . . 5 {⟨∅, 1𝑜⟩} ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
11 opex 5074 . . . . . 6 ⟨∅, 1𝑜⟩ ∈ V
1211snss 4462 . . . . 5 (⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ↔ {⟨∅, 1𝑜⟩} ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
1310, 12mpbir 222 . . . 4 ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
14 relxp 5280 . . . . . . . 8 Rel (V × V)
15 cdafn 9214 . . . . . . . . . 10 +𝑐 Fn (V × V)
16 fndm 6141 . . . . . . . . . 10 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
1715, 16ax-mp 5 . . . . . . . . 9 dom +𝑐 = (V × V)
1817releqi 5354 . . . . . . . 8 (Rel dom +𝑐 ↔ Rel (V × V))
1914, 18mpbir 222 . . . . . . 7 Rel dom +𝑐
2019ovrcl 6852 . . . . . 6 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ V ∧ 1𝑜 ∈ V))
2120simpld 483 . . . . 5 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
22 1on 7741 . . . . 5 1𝑜 ∈ On
23 cdaval 9215 . . . . 5 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
2421, 22, 23sylancl 575 . . . 4 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
2513, 24syl5eleqr 2860 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜))
26 difsnen 8219 . . 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 1480 . 2 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}))
2824difeq1d 3885 . . . 4 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩}))
29 xp01disj 7751 . . . . . 6 ((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
30 disj3 4174 . . . . . 6 (((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜})))
3129, 30mpbi 221 . . . . 5 (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
32 difun2 4200 . . . . 5 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
338difeq2i 3883 . . . . 5 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩})
3431, 32, 333eqtr2i 2802 . . . 4 (𝐴 × {∅}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩})
3528, 34syl6eqr 2826 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (𝐴 × {∅}))
36 xpsneng 8222 . . . 4 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
3721, 5, 36sylancl 575 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ 𝐴)
3835, 37eqbrtrd 4819 . 2 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ 𝐴)
39 entr 8182 . 2 ((((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ∧ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ 𝐴) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)
4027, 38, 39syl2anc 574 1 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1634  wcel 2148  Vcvv 3355  cdif 3726  cun 3727  cin 3728  wss 3729  c0 4073  {csn 4326  cop 4332   class class class wbr 4797   × cxp 5261  dom cdm 5263  Rel wrel 5268  Oncon0 5877   Fn wfn 6037  (class class class)co 6812  1𝑜c1o 7727  cen 8127   +𝑐 ccda 9212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-int 4623  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-ord 5880  df-on 5881  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-1st 7336  df-2nd 7337  df-1o 7734  df-er 7917  df-en 8131  df-cda 9213
This theorem is referenced by:  canthp1  9699
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