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Theorem infcda1 8960
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 7906 . . . . . . . 8 Rel ≼
21brrelex2i 5124 . . . . . . 7 (ω ≼ 𝐴𝐴 ∈ V)
3 1on 7513 . . . . . . 7 1𝑜 ∈ On
4 cdaval 8937 . . . . . . 7 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
52, 3, 4sylancl 693 . . . . . 6 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
6 df1o2 7518 . . . . . . . . 9 1𝑜 = {∅}
76xpeq1i 5100 . . . . . . . 8 (1𝑜 × {1𝑜}) = ({∅} × {1𝑜})
8 0ex 4755 . . . . . . . . 9 ∅ ∈ V
93elexi 3204 . . . . . . . . 9 1𝑜 ∈ V
108, 9xpsn 6362 . . . . . . . 8 ({∅} × {1𝑜}) = {⟨∅, 1𝑜⟩}
117, 10eqtr2i 2649 . . . . . . 7 {⟨∅, 1𝑜⟩} = (1𝑜 × {1𝑜})
1211a1i 11 . . . . . 6 (ω ≼ 𝐴 → {⟨∅, 1𝑜⟩} = (1𝑜 × {1𝑜}))
135, 12difeq12d 3712 . . . . 5 (ω ≼ 𝐴 → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})))
14 difun2 4025 . . . . . 6 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
15 xp01disj 7522 . . . . . . 7 ((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
16 disj3 3998 . . . . . . 7 (((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜})))
1715, 16mpbi 220 . . . . . 6 (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
1814, 17eqtr4i 2651 . . . . 5 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = (𝐴 × {∅})
1913, 18syl6eq 2676 . . . 4 (ω ≼ 𝐴 → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (𝐴 × {∅}))
20 cdadom3 8955 . . . . . . 7 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
212, 3, 20sylancl 693 . . . . . 6 (ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 1𝑜))
22 domtr 7954 . . . . . 6 ((ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 1𝑜)) → ω ≼ (𝐴 +𝑐 1𝑜))
2321, 22mpdan 701 . . . . 5 (ω ≼ 𝐴 → ω ≼ (𝐴 +𝑐 1𝑜))
24 infdifsn 8499 . . . . 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 4642 . . 3 (ω ≼ 𝐴 → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
2726ensymd 7952 . 2 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ (𝐴 × {∅}))
28 xpsneng 7990 . . 3 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
292, 8, 28sylancl 693 . 2 (ω ≼ 𝐴 → (𝐴 × {∅}) ≈ 𝐴)
30 entr 7953 . 2 (((𝐴 +𝑐 1𝑜) ≈ (𝐴 × {∅}) ∧ (𝐴 × {∅}) ≈ 𝐴) → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
3127, 29, 30syl2anc 692 1 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  Vcvv 3191  cdif 3557  cun 3558  cin 3559  c0 3896  {csn 4153  cop 4159   class class class wbr 4618   × cxp 5077  Oncon0 5685  (class class class)co 6605  ωcom 7013  1𝑜c1o 7499  cen 7897  cdom 7898   +𝑐 ccda 8934
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  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-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 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1o 7506  df-er 7688  df-en 7901  df-dom 7902  df-cda 8935
This theorem is referenced by:  pwcdaidm  8962  isfin4-3  9082  canthp1lem2  9420
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