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Theorem infcda1 9300
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 8198 . . . . . . . 8 Rel ≼
21brrelex2i 5359 . . . . . . 7 (ω ≼ 𝐴𝐴 ∈ V)
3 1on 7803 . . . . . . 7 1𝑜 ∈ On
4 cdaval 9277 . . . . . . 7 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
52, 3, 4sylancl 576 . . . . . 6 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
6 df1o2 7809 . . . . . . . . 9 1𝑜 = {∅}
76xpeq1i 5336 . . . . . . . 8 (1𝑜 × {1𝑜}) = ({∅} × {1𝑜})
8 0ex 4984 . . . . . . . . 9 ∅ ∈ V
9 1oex 7804 . . . . . . . . 9 1𝑜 ∈ V
108, 9xpsn 6630 . . . . . . . 8 ({∅} × {1𝑜}) = {⟨∅, 1𝑜⟩}
117, 10eqtr2i 2829 . . . . . . 7 {⟨∅, 1𝑜⟩} = (1𝑜 × {1𝑜})
1211a1i 11 . . . . . 6 (ω ≼ 𝐴 → {⟨∅, 1𝑜⟩} = (1𝑜 × {1𝑜}))
135, 12difeq12d 3928 . . . . 5 (ω ≼ 𝐴 → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})))
14 difun2 4244 . . . . . 6 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
15 xp01disj 7813 . . . . . . 7 ((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅
16 disj3 4218 . . . . . . 7 (((𝐴 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜})))
1715, 16mpbi 221 . . . . . 6 (𝐴 × {∅}) = ((𝐴 × {∅}) ∖ (1𝑜 × {1𝑜}))
1814, 17eqtr4i 2831 . . . . 5 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ∖ (1𝑜 × {1𝑜})) = (𝐴 × {∅})
1913, 18syl6eq 2856 . . . 4 (ω ≼ 𝐴 → ((𝐴 +𝑐 1𝑜) ∖ {⟨∅, 1𝑜⟩}) = (𝐴 × {∅}))
20 cdadom3 9295 . . . . . . 7 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → 𝐴 ≼ (𝐴 +𝑐 1𝑜))
212, 3, 20sylancl 576 . . . . . 6 (ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 1𝑜))
22 domtr 8245 . . . . . 6 ((ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 1𝑜)) → ω ≼ (𝐴 +𝑐 1𝑜))
2321, 22mpdan 670 . . . . 5 (ω ≼ 𝐴 → ω ≼ (𝐴 +𝑐 1𝑜))
24 infdifsn 8801 . . . . 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 4868 . . 3 (ω ≼ 𝐴 → (𝐴 × {∅}) ≈ (𝐴 +𝑐 1𝑜))
2726ensymd 8243 . 2 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ (𝐴 × {∅}))
28 xpsneng 8284 . . 3 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
292, 8, 28sylancl 576 . 2 (ω ≼ 𝐴 → (𝐴 × {∅}) ≈ 𝐴)
30 entr 8244 . 2 (((𝐴 +𝑐 1𝑜) ≈ (𝐴 × {∅}) ∧ (𝐴 × {∅}) ≈ 𝐴) → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
3127, 29, 30syl2anc 575 1 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2156  Vcvv 3391  cdif 3766  cun 3767  cin 3768  c0 4116  {csn 4370  cop 4376   class class class wbr 4844   × cxp 5309  Oncon0 5936  (class class class)co 6874  ωcom 7295  1𝑜c1o 7789  cen 8189  cdom 8190   +𝑐 ccda 9274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1o 7796  df-er 7979  df-en 8193  df-dom 8194  df-cda 9275
This theorem is referenced by:  pwcdaidm  9302  isfin4-3  9422  canthp1lem2  9760
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