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Theorem cda1dif 8881
 Description: Adding and subtracting one gives back the original set. Similar to pncan 10166 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 ovex 6577 . . . 4 (𝐴 +𝑐 1𝑜) ∈ V
21a1i 11 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) ∈ V)
3 id 22 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → 𝐵 ∈ (𝐴 +𝑐 1𝑜))
4 df1o2 7459 . . . . . . . 8 1𝑜 = {∅}
54xpeq1i 5059 . . . . . . 7 (1𝑜 × {1𝑜}) = ({∅} × {1𝑜})
6 0ex 4718 . . . . . . . 8 ∅ ∈ V
7 1on 7454 . . . . . . . . 9 1𝑜 ∈ On
87elexi 3186 . . . . . . . 8 1𝑜 ∈ V
96, 8xpsn 6313 . . . . . . 7 ({∅} × {1𝑜}) = {⟨∅, 1𝑜⟩}
105, 9eqtri 2632 . . . . . 6 (1𝑜 × {1𝑜}) = {⟨∅, 1𝑜⟩}
11 ssun2 3739 . . . . . 6 (1𝑜 × {1𝑜}) ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
1210, 11eqsstr3i 3599 . . . . 5 {⟨∅, 1𝑜⟩} ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
13 opex 4859 . . . . . 6 ⟨∅, 1𝑜⟩ ∈ V
1413snss 4259 . . . . 5 (⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ↔ {⟨∅, 1𝑜⟩} ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
1512, 14mpbir 220 . . . 4 ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
16 relxp 5150 . . . . . . . 8 Rel (V × V)
17 cdafn 8874 . . . . . . . . . 10 +𝑐 Fn (V × V)
18 fndm 5904 . . . . . . . . . 10 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
1917, 18ax-mp 5 . . . . . . . . 9 dom +𝑐 = (V × V)
2019releqi 5125 . . . . . . . 8 (Rel dom +𝑐 ↔ Rel (V × V))
2116, 20mpbir 220 . . . . . . 7 Rel dom +𝑐
2221ovrcl 6584 . . . . . 6 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ V ∧ 1𝑜 ∈ V))
2322simpld 474 . . . . 5 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
24 cdaval 8875 . . . . 5 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
2523, 7, 24sylancl 693 . . . 4 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
2615, 25syl5eleqr 2695 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜))
27 difsnen 7927 . . 3 (((𝐴 +𝑐 1𝑜) ∈ V ∧ 𝐵 ∈ (𝐴 +𝑐 1𝑜) ∧ ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜)) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}))
282, 3, 26, 27syl3anc 1318 . 2 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}))
2925difeq1d 3689 . . . 4 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩}))
30 xp01disj 7463 . . . . . 6 ((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
31 disj3 3973 . . . . . 6 (((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜})))
3230, 31mpbi 219 . . . . 5 (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
33 difun2 4000 . . . . 5 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
3410difeq2i 3687 . . . . 5 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩})
3532, 33, 343eqtr2i 2638 . . . 4 (𝐴 × {∅}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩})
3629, 35syl6eqr 2662 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (𝐴 × {∅}))
37 xpsneng 7930 . . . 4 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
3823, 6, 37sylancl 693 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ 𝐴)
3936, 38eqbrtrd 4605 . 2 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ 𝐴)
40 entr 7894 . 2 ((((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ∧ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ 𝐴) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)
4128, 39, 40syl2anc 691 1 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583   × cxp 5036  dom cdm 5038  Rel wrel 5043  Oncon0 5640   Fn wfn 5799  (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-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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-iun 4457  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-1st 7059  df-2nd 7060  df-1o 7447  df-er 7629  df-en 7842  df-cda 8873 This theorem is referenced by:  canthp1  9355
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