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Theorem infcda1 8898
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 7847 . . . . . . . 8 Rel ≼
21brrelex2i 5083 . . . . . . 7 (ω ≼ 𝐴𝐴 ∈ V)
3 1on 7454 . . . . . . 7 1𝑜 ∈ On
4 cdaval 8875 . . . . . . 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 7459 . . . . . . . . 9 1𝑜 = {∅}
76xpeq1i 5059 . . . . . . . 8 (1𝑜 × {1𝑜}) = ({∅} × {1𝑜})
8 0ex 4718 . . . . . . . . 9 ∅ ∈ V
93elexi 3186 . . . . . . . . 9 1𝑜 ∈ V
108, 9xpsn 6313 . . . . . . . 8 ({∅} × {1𝑜}) = {⟨∅, 1𝑜⟩}
117, 10eqtr2i 2633 . . . . . . 7 {⟨∅, 1𝑜⟩} = (1𝑜 × {1𝑜})
1211a1i 11 . . . . . 6 (ω ≼ 𝐴 → {⟨∅, 1𝑜⟩} = (1𝑜 × {1𝑜}))
135, 12difeq12d 3691 . . . . 5 (ω ≼ 𝐴 → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})))
14 difun2 4000 . . . . . 6 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
15 xp01disj 7463 . . . . . . 7 ((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
16 disj3 3973 . . . . . . 7 (((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜})))
1715, 16mpbi 219 . . . . . 6 (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
1814, 17eqtr4i 2635 . . . . 5 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = (𝐴 × {∅})
1913, 18syl6eq 2660 . . . 4 (ω ≼ 𝐴 → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (𝐴 × {∅}))
20 cdadom3 8893 . . . . . . 7 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
212, 3, 20sylancl 693 . . . . . 6 (ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 1𝑜))
22 domtr 7895 . . . . . 6 ((ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 1𝑜)) → ω ≼ (𝐴 +𝑐 1𝑜))
2321, 22mpdan 699 . . . . 5 (ω ≼ 𝐴 → ω ≼ (𝐴 +𝑐 1𝑜))
24 infdifsn 8437 . . . . 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 4607 . . 3 (ω ≼ 𝐴 → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
2726ensymd 7893 . 2 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ (𝐴 × {∅}))
28 xpsneng 7930 . . 3 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
292, 8, 28sylancl 693 . 2 (ω ≼ 𝐴 → (𝐴 × {∅}) ≈ 𝐴)
30 entr 7894 . 2 (((𝐴 +𝑐 1𝑜) ≈ (𝐴 × {∅}) ∧ (𝐴 × {∅}) ≈ 𝐴) → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
3127, 29, 30syl2anc 691 1 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  Vcvv 3173  cdif 3537  cun 3538  cin 3539  c0 3874  {csn 4125  cop 4131   class class class wbr 4583   × cxp 5036  Oncon0 5640  (class class class)co 6549  ωcom 6957  1𝑜c1o 7440  cen 7838  cdom 7839   +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-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-lim 5645  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-om 6958  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-cda 8873
This theorem is referenced by:  pwcdaidm  8900  isfin4-3  9020  canthp1lem2  9354
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