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Theorem infcda1 9053
Description: An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
infcda1 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)

Proof of Theorem infcda1
StepHypRef Expression
1 reldom 8003 . . . . . . . 8 Rel ≼
21brrelex2i 5193 . . . . . . 7 (ω ≼ 𝐴𝐴 ∈ V)
3 1on 7612 . . . . . . 7 1𝑜 ∈ On
4 cdaval 9030 . . . . . . 7 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
52, 3, 4sylancl 695 . . . . . 6 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
6 df1o2 7617 . . . . . . . . 9 1𝑜 = {∅}
76xpeq1i 5169 . . . . . . . 8 (1𝑜 × {1𝑜}) = ({∅} × {1𝑜})
8 0ex 4823 . . . . . . . . 9 ∅ ∈ V
93elexi 3244 . . . . . . . . 9 1𝑜 ∈ V
108, 9xpsn 6447 . . . . . . . 8 ({∅} × {1𝑜}) = {⟨∅, 1𝑜⟩}
117, 10eqtr2i 2674 . . . . . . 7 {⟨∅, 1𝑜⟩} = (1𝑜 × {1𝑜})
1211a1i 11 . . . . . 6 (ω ≼ 𝐴 → {⟨∅, 1𝑜⟩} = (1𝑜 × {1𝑜}))
135, 12difeq12d 3762 . . . . 5 (ω ≼ 𝐴 → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})))
14 difun2 4081 . . . . . 6 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
15 xp01disj 7621 . . . . . . 7 ((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
16 disj3 4054 . . . . . . 7 (((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜})))
1715, 16mpbi 220 . . . . . 6 (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
1814, 17eqtr4i 2676 . . . . 5 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = (𝐴 × {∅})
1913, 18syl6eq 2701 . . . 4 (ω ≼ 𝐴 → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (𝐴 × {∅}))
20 cdadom3 9048 . . . . . . 7 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
212, 3, 20sylancl 695 . . . . . 6 (ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 1𝑜))
22 domtr 8050 . . . . . 6 ((ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 1𝑜)) → ω ≼ (𝐴 +𝑐 1𝑜))
2321, 22mpdan 703 . . . . 5 (ω ≼ 𝐴 → ω ≼ (𝐴 +𝑐 1𝑜))
24 infdifsn 8592 . . . . 5 (ω ≼ (𝐴 +𝑐 1𝑜) → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ (𝐴 +𝑐 1𝑜))
2523, 24syl 17 . . . 4 (ω ≼ 𝐴 → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) ≈ (𝐴 +𝑐 1𝑜))
2619, 25eqbrtrrd 4709 . . 3 (ω ≼ 𝐴 → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
2726ensymd 8048 . 2 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ (𝐴 × {∅}))
28 xpsneng 8086 . . 3 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
292, 8, 28sylancl 695 . 2 (ω ≼ 𝐴 → (𝐴 × {∅}) ≈ 𝐴)
30 entr 8049 . 2 (((𝐴 +𝑐 1𝑜) ≈ (𝐴 × {∅}) ∧ (𝐴 × {∅}) ≈ 𝐴) → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
3127, 29, 30syl2anc 694 1 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cdif 3604  cun 3605  cin 3606  c0 3948  {csn 4210  cop 4216   class class class wbr 4685   × cxp 5141  Oncon0 5761  (class class class)co 6690  ωcom 7107  1𝑜c1o 7598  cen 7994  cdom 7995   +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-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-lim 5766  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-om 7108  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-cda 9028
This theorem is referenced by:  pwcdaidm  9055  isfin4-3  9175  canthp1lem2  9513
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