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Theorem cda1dif 8949
Description: Adding and subtracting one gives back the original set. Similar to pncan 10238 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 6638 . . . 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 7524 . . . . . . . 8 1𝑜 = {∅}
54xpeq1i 5100 . . . . . . 7 (1𝑜 × {1𝑜}) = ({∅} × {1𝑜})
6 0ex 4755 . . . . . . . 8 ∅ ∈ V
7 1on 7519 . . . . . . . . 9 1𝑜 ∈ On
87elexi 3202 . . . . . . . 8 1𝑜 ∈ V
96, 8xpsn 6367 . . . . . . 7 ({∅} × {1𝑜}) = {⟨∅, 1𝑜⟩}
105, 9eqtri 2643 . . . . . 6 (1𝑜 × {1𝑜}) = {⟨∅, 1𝑜⟩}
11 ssun2 3760 . . . . . 6 (1𝑜 × {1𝑜}) ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
1210, 11eqsstr3i 3620 . . . . 5 {⟨∅, 1𝑜⟩} ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
13 opex 4898 . . . . . 6 ⟨∅, 1𝑜⟩ ∈ V
1413snss 4291 . . . . 5 (⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ↔ {⟨∅, 1𝑜⟩} ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
1512, 14mpbir 221 . . . 4 ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
16 relxp 5193 . . . . . . . 8 Rel (V × V)
17 cdafn 8942 . . . . . . . . . 10 +𝑐 Fn (V × V)
18 fndm 5953 . . . . . . . . . 10 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
1917, 18ax-mp 5 . . . . . . . . 9 dom +𝑐 = (V × V)
2019releqi 5168 . . . . . . . 8 (Rel dom +𝑐 ↔ Rel (V × V))
2116, 20mpbir 221 . . . . . . 7 Rel dom +𝑐
2221ovrcl 6646 . . . . . 6 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ V ∧ 1𝑜 ∈ V))
2322simpld 475 . . . . 5 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
24 cdaval 8943 . . . . 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 2705 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜))
27 difsnen 7993 . . 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 1323 . 2 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}))
2925difeq1d 3710 . . . 4 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩}))
30 xp01disj 7528 . . . . . 6 ((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
31 disj3 3998 . . . . . 6 (((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜})))
3230, 31mpbi 220 . . . . 5 (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
33 difun2 4025 . . . . 5 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
3410difeq2i 3708 . . . . 5 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩})
3532, 33, 343eqtr2i 2649 . . . 4 (𝐴 × {∅}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩})
3629, 35syl6eqr 2673 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (𝐴 × {∅}))
37 xpsneng 7996 . . . 4 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
3823, 6, 37sylancl 693 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ 𝐴)
3936, 38eqbrtrd 4640 . 2 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ 𝐴)
40 entr 7959 . 2 ((((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ∧ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ 𝐴) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)
4128, 39, 40syl2anc 692 1 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3189  cdif 3556  cun 3557  cin 3558  wss 3559  c0 3896  {csn 4153  cop 4159   class class class wbr 4618   × cxp 5077  dom cdm 5079  Rel wrel 5084  Oncon0 5687   Fn wfn 5847  (class class class)co 6610  1𝑜c1o 7505  cen 7903   +𝑐 ccda 8940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-1o 7512  df-er 7694  df-en 7907  df-cda 8941
This theorem is referenced by:  canthp1  9427
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