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Theorem cdainf 9266
Description: A set is infinite iff the cardinal sum with itself is infinite. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
cdainf (ω ≼ 𝐴 ↔ ω ≼ (𝐴 +𝑐 𝐴))

Proof of Theorem cdainf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 reldom 8165 . . . . 5 Rel ≼
21brrelex2i 5328 . . . 4 (ω ≼ 𝐴𝐴 ∈ V)
3 cdadom3 9262 . . . 4 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → 𝐴 ≼ (𝐴 +𝑐 𝐴))
42, 2, 3syl2anc 579 . . 3 (ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 𝐴))
5 domtr 8212 . . 3 ((ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 𝐴)) → ω ≼ (𝐴 +𝑐 𝐴))
64, 5mpdan 678 . 2 (ω ≼ 𝐴 → ω ≼ (𝐴 +𝑐 𝐴))
7 infn0 8428 . . . 4 (ω ≼ (𝐴 +𝑐 𝐴) → (𝐴 +𝑐 𝐴) ≠ ∅)
8 cdafn 9243 . . . . . . . 8 +𝑐 Fn (V × V)
9 fndm 6167 . . . . . . . 8 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
108, 9ax-mp 5 . . . . . . 7 dom +𝑐 = (V × V)
1110ndmov 7015 . . . . . 6 (¬ (𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 +𝑐 𝐴) = ∅)
1211necon1ai 2963 . . . . 5 ((𝐴 +𝑐 𝐴) ≠ ∅ → (𝐴 ∈ V ∧ 𝐴 ∈ V))
1312simpld 488 . . . 4 ((𝐴 +𝑐 𝐴) ≠ ∅ → 𝐴 ∈ V)
147, 13syl 17 . . 3 (ω ≼ (𝐴 +𝑐 𝐴) → 𝐴 ∈ V)
15 ovex 6873 . . . . 5 (𝐴 +𝑐 𝐴) ∈ V
1615domen 8172 . . . 4 (ω ≼ (𝐴 +𝑐 𝐴) ↔ ∃𝑥(ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴)))
17 indi 4037 . . . . . . . . 9 (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) = ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜})))
18 simprr 789 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ⊆ (𝐴 +𝑐 𝐴))
19 simpl 474 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝐴 ∈ V)
20 cdaval 9244 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
2119, 19, 20syl2anc 579 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
2218, 21sseqtrd 3800 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ⊆ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
23 df-ss 3745 . . . . . . . . . 10 (𝑥 ⊆ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})) ↔ (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) = 𝑥)
2422, 23sylib 209 . . . . . . . . 9 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) = 𝑥)
2517, 24syl5eqr 2812 . . . . . . . 8 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) = 𝑥)
26 ensym 8208 . . . . . . . . 9 (ω ≈ 𝑥𝑥 ≈ ω)
2726ad2antrl 719 . . . . . . . 8 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ≈ ω)
2825, 27eqbrtrd 4830 . . . . . . 7 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω)
2928ex 401 . . . . . 6 (𝐴 ∈ V → ((ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴)) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω))
30 cdainflem 9265 . . . . . . 7 (((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω → ((𝑥 ∩ (𝐴 × {∅})) ≈ ω ∨ (𝑥 ∩ (𝐴 × {1𝑜})) ≈ ω))
31 snex 5063 . . . . . . . . . . . 12 {∅} ∈ V
32 xpexg 7157 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
3331, 32mpan2 682 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 × {∅}) ∈ V)
34 inss2 3992 . . . . . . . . . . 11 (𝑥 ∩ (𝐴 × {∅})) ⊆ (𝐴 × {∅})
35 ssdomg 8205 . . . . . . . . . . 11 ((𝐴 × {∅}) ∈ V → ((𝑥 ∩ (𝐴 × {∅})) ⊆ (𝐴 × {∅}) → (𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅})))
3633, 34, 35mpisyl 21 . . . . . . . . . 10 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅}))
37 0ex 4949 . . . . . . . . . . 11 ∅ ∈ V
38 xpsneng 8251 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
3937, 38mpan2 682 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 × {∅}) ≈ 𝐴)
40 domentr 8218 . . . . . . . . . 10 (((𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅}) ∧ (𝐴 × {∅}) ≈ 𝐴) → (𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴)
4136, 39, 40syl2anc 579 . . . . . . . . 9 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴)
42 domen1 8308 . . . . . . . . 9 ((𝑥 ∩ (𝐴 × {∅})) ≈ ω → ((𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴 ↔ ω ≼ 𝐴))
4341, 42syl5ibcom 236 . . . . . . . 8 (𝐴 ∈ V → ((𝑥 ∩ (𝐴 × {∅})) ≈ ω → ω ≼ 𝐴))
44 snex 5063 . . . . . . . . . . . 12 {1𝑜} ∈ V
45 xpexg 7157 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ {1𝑜} ∈ V) → (𝐴 × {1𝑜}) ∈ V)
4644, 45mpan2 682 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 × {1𝑜}) ∈ V)
47 inss2 3992 . . . . . . . . . . 11 (𝑥 ∩ (𝐴 × {1𝑜})) ⊆ (𝐴 × {1𝑜})
48 ssdomg 8205 . . . . . . . . . . 11 ((𝐴 × {1𝑜}) ∈ V → ((𝑥 ∩ (𝐴 × {1𝑜})) ⊆ (𝐴 × {1𝑜}) → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ (𝐴 × {1𝑜})))
4946, 47, 48mpisyl 21 . . . . . . . . . 10 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ (𝐴 × {1𝑜}))
50 1on 7770 . . . . . . . . . . 11 1𝑜 ∈ On
51 xpsneng 8251 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
5250, 51mpan2 682 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 × {1𝑜}) ≈ 𝐴)
53 domentr 8218 . . . . . . . . . 10 (((𝑥 ∩ (𝐴 × {1𝑜})) ≼ (𝐴 × {1𝑜}) ∧ (𝐴 × {1𝑜}) ≈ 𝐴) → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴)
5449, 52, 53syl2anc 579 . . . . . . . . 9 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴)
55 domen1 8308 . . . . . . . . 9 ((𝑥 ∩ (𝐴 × {1𝑜})) ≈ ω → ((𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴 ↔ ω ≼ 𝐴))
5654, 55syl5ibcom 236 . . . . . . . 8 (𝐴 ∈ V → ((𝑥 ∩ (𝐴 × {1𝑜})) ≈ ω → ω ≼ 𝐴))
5743, 56jaod 885 . . . . . . 7 (𝐴 ∈ V → (((𝑥 ∩ (𝐴 × {∅})) ≈ ω ∨ (𝑥 ∩ (𝐴 × {1𝑜})) ≈ ω) → ω ≼ 𝐴))
5830, 57syl5 34 . . . . . 6 (𝐴 ∈ V → (((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω → ω ≼ 𝐴))
5929, 58syld 47 . . . . 5 (𝐴 ∈ V → ((ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴)) → ω ≼ 𝐴))
6059exlimdv 2028 . . . 4 (𝐴 ∈ V → (∃𝑥(ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴)) → ω ≼ 𝐴))
6116, 60syl5bi 233 . . 3 (𝐴 ∈ V → (ω ≼ (𝐴 +𝑐 𝐴) → ω ≼ 𝐴))
6214, 61mpcom 38 . 2 (ω ≼ (𝐴 +𝑐 𝐴) → ω ≼ 𝐴)
636, 62impbii 200 1 (ω ≼ 𝐴 ↔ ω ≼ (𝐴 +𝑐 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wo 873   = wceq 1652  wex 1874  wcel 2155  wne 2936  Vcvv 3349  cun 3729  cin 3730  wss 3731  c0 4078  {csn 4333   class class class wbr 4808   × cxp 5274  dom cdm 5276  Oncon0 5907   Fn wfn 6062  (class class class)co 6841  ωcom 7262  1𝑜c1o 7756  cen 8156  cdom 8157   +𝑐 ccda 9241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-oadd 7767  df-er 7946  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-cda 9242
This theorem is referenced by:  infdif  9283
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