MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cda1dif Structured version   Visualization version   GIF version

Theorem cda1dif 9203
Description: Adding and subtracting one gives back the original set. Similar to pncan 10492 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 6828 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) ∈ V)
2 id 22 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → 𝐵 ∈ (𝐴 +𝑐 1𝑜))
3 df1o2 7729 . . . . . . . 8 1𝑜 = {∅}
43xpeq1i 5275 . . . . . . 7 (1𝑜 × {1𝑜}) = ({∅} × {1𝑜})
5 0ex 4925 . . . . . . . 8 ∅ ∈ V
6 1oex 7724 . . . . . . . 8 1𝑜 ∈ V
75, 6xpsn 6552 . . . . . . 7 ({∅} × {1𝑜}) = {⟨∅, 1𝑜⟩}
84, 7eqtri 2793 . . . . . 6 (1𝑜 × {1𝑜}) = {⟨∅, 1𝑜⟩}
9 ssun2 3928 . . . . . 6 (1𝑜 × {1𝑜}) ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
108, 9eqsstr3i 3785 . . . . 5 {⟨∅, 1𝑜⟩} ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
11 opex 5061 . . . . . 6 ⟨∅, 1𝑜⟩ ∈ V
1211snss 4452 . . . . 5 (⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ↔ {⟨∅, 1𝑜⟩} ⊆ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
1310, 12mpbir 221 . . . 4 ⟨∅, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
14 relxp 5267 . . . . . . . 8 Rel (V × V)
15 cdafn 9196 . . . . . . . . . 10 +𝑐 Fn (V × V)
16 fndm 6129 . . . . . . . . . 10 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
1715, 16ax-mp 5 . . . . . . . . 9 dom +𝑐 = (V × V)
1817releqi 5341 . . . . . . . 8 (Rel dom +𝑐 ↔ Rel (V × V))
1914, 18mpbir 221 . . . . . . 7 Rel dom +𝑐
2019ovrcl 6834 . . . . . 6 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 ∈ V ∧ 1𝑜 ∈ V))
2120simpld 482 . . . . 5 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → 𝐴 ∈ V)
22 1on 7723 . . . . 5 1𝑜 ∈ On
23 cdaval 9197 . . . . 5 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
2421, 22, 23sylancl 574 . . . 4 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
2513, 24syl5eleqr 2857 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ⟨∅, 1𝑜⟩ ∈ (𝐴 +𝑐 1𝑜))
26 difsnen 8201 . . 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 1476 . 2 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}))
2824difeq1d 3878 . . . 4 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩}))
29 xp01disj 7733 . . . . . 6 ((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
30 disj3 4165 . . . . . 6 (((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜})))
3129, 30mpbi 220 . . . . 5 (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
32 difun2 4191 . . . . 5 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
338difeq2i 3876 . . . . 5 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩})
3431, 32, 333eqtr2i 2799 . . . 4 (𝐴 × {∅}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ {⟨∅, 1𝑜⟩})
3528, 34syl6eqr 2823 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (𝐴 × {∅}))
36 xpsneng 8204 . . . 4 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
3721, 5, 36sylancl 574 . . 3 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → (𝐴 × {∅}) ≈ 𝐴)
3835, 37eqbrtrd 4809 . 2 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ 𝐴)
39 entr 8164 . 2 ((((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ∧ ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ 𝐴) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)
4027, 38, 39syl2anc 573 1 (𝐵 ∈ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {𝐵}) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  Vcvv 3351  cdif 3720  cun 3721  cin 3722  wss 3723  c0 4063  {csn 4317  cop 4323   class class class wbr 4787   × cxp 5248  dom cdm 5250  Rel wrel 5255  Oncon0 5865   Fn wfn 6025  (class class class)co 6795  1𝑜c1o 7709  cen 8109   +𝑐 ccda 9194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-ord 5868  df-on 5869  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-1st 7318  df-2nd 7319  df-1o 7716  df-er 7899  df-en 8113  df-cda 9195
This theorem is referenced by:  canthp1  9681
  Copyright terms: Public domain W3C validator