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Theorem cdainf 9052
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 8003 . . . . 5 Rel ≼
21brrelex2i 5193 . . . 4 (ω ≼ 𝐴𝐴 ∈ V)
3 cdadom3 9048 . . . 4 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → 𝐴 ≼ (𝐴 +𝑐 𝐴))
42, 2, 3syl2anc 694 . . 3 (ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 𝐴))
5 domtr 8050 . . 3 ((ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 𝐴)) → ω ≼ (𝐴 +𝑐 𝐴))
64, 5mpdan 703 . 2 (ω ≼ 𝐴 → ω ≼ (𝐴 +𝑐 𝐴))
7 infn0 8263 . . . 4 (ω ≼ (𝐴 +𝑐 𝐴) → (𝐴 +𝑐 𝐴) ≠ ∅)
8 cdafn 9029 . . . . . . . 8 +𝑐 Fn (V × V)
9 fndm 6028 . . . . . . . 8 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
108, 9ax-mp 5 . . . . . . 7 dom +𝑐 = (V × V)
1110ndmov 6860 . . . . . 6 (¬ (𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 +𝑐 𝐴) = ∅)
1211necon1ai 2850 . . . . 5 ((𝐴 +𝑐 𝐴) ≠ ∅ → (𝐴 ∈ V ∧ 𝐴 ∈ V))
1312simpld 474 . . . 4 ((𝐴 +𝑐 𝐴) ≠ ∅ → 𝐴 ∈ V)
147, 13syl 17 . . 3 (ω ≼ (𝐴 +𝑐 𝐴) → 𝐴 ∈ V)
15 ovex 6718 . . . . 5 (𝐴 +𝑐 𝐴) ∈ V
1615domen 8010 . . . 4 (ω ≼ (𝐴 +𝑐 𝐴) ↔ ∃𝑥(ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴)))
17 indi 3906 . . . . . . . . 9 (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) = ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜})))
18 simprr 811 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ⊆ (𝐴 +𝑐 𝐴))
19 simpl 472 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝐴 ∈ V)
20 cdaval 9030 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
2119, 19, 20syl2anc 694 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
2218, 21sseqtrd 3674 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ⊆ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
23 df-ss 3621 . . . . . . . . . 10 (𝑥 ⊆ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})) ↔ (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) = 𝑥)
2422, 23sylib 208 . . . . . . . . 9 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) = 𝑥)
2517, 24syl5eqr 2699 . . . . . . . 8 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) = 𝑥)
26 ensym 8046 . . . . . . . . 9 (ω ≈ 𝑥𝑥 ≈ ω)
2726ad2antrl 764 . . . . . . . 8 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ≈ ω)
2825, 27eqbrtrd 4707 . . . . . . 7 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω)
2928ex 449 . . . . . 6 (𝐴 ∈ V → ((ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴)) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω))
30 cdainflem 9051 . . . . . . 7 (((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω → ((𝑥 ∩ (𝐴 × {∅})) ≈ ω ∨ (𝑥 ∩ (𝐴 × {1𝑜})) ≈ ω))
31 snex 4938 . . . . . . . . . . . 12 {∅} ∈ V
32 xpexg 7002 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
3331, 32mpan2 707 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 × {∅}) ∈ V)
34 inss2 3867 . . . . . . . . . . 11 (𝑥 ∩ (𝐴 × {∅})) ⊆ (𝐴 × {∅})
35 ssdomg 8043 . . . . . . . . . . 11 ((𝐴 × {∅}) ∈ V → ((𝑥 ∩ (𝐴 × {∅})) ⊆ (𝐴 × {∅}) → (𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅})))
3633, 34, 35mpisyl 21 . . . . . . . . . 10 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅}))
37 0ex 4823 . . . . . . . . . . 11 ∅ ∈ V
38 xpsneng 8086 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
3937, 38mpan2 707 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 × {∅}) ≈ 𝐴)
40 domentr 8056 . . . . . . . . . 10 (((𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅}) ∧ (𝐴 × {∅}) ≈ 𝐴) → (𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴)
4136, 39, 40syl2anc 694 . . . . . . . . 9 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴)
42 domen1 8143 . . . . . . . . 9 ((𝑥 ∩ (𝐴 × {∅})) ≈ ω → ((𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴 ↔ ω ≼ 𝐴))
4341, 42syl5ibcom 235 . . . . . . . 8 (𝐴 ∈ V → ((𝑥 ∩ (𝐴 × {∅})) ≈ ω → ω ≼ 𝐴))
44 snex 4938 . . . . . . . . . . . 12 {1𝑜} ∈ V
45 xpexg 7002 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ {1𝑜} ∈ V) → (𝐴 × {1𝑜}) ∈ V)
4644, 45mpan2 707 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 × {1𝑜}) ∈ V)
47 inss2 3867 . . . . . . . . . . 11 (𝑥 ∩ (𝐴 × {1𝑜})) ⊆ (𝐴 × {1𝑜})
48 ssdomg 8043 . . . . . . . . . . 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 7612 . . . . . . . . . . 11 1𝑜 ∈ On
51 xpsneng 8086 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
5250, 51mpan2 707 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 × {1𝑜}) ≈ 𝐴)
53 domentr 8056 . . . . . . . . . 10 (((𝑥 ∩ (𝐴 × {1𝑜})) ≼ (𝐴 × {1𝑜}) ∧ (𝐴 × {1𝑜}) ≈ 𝐴) → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴)
5449, 52, 53syl2anc 694 . . . . . . . . 9 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴)
55 domen1 8143 . . . . . . . . 9 ((𝑥 ∩ (𝐴 × {1𝑜})) ≈ ω → ((𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴 ↔ ω ≼ 𝐴))
5654, 55syl5ibcom 235 . . . . . . . 8 (𝐴 ∈ V → ((𝑥 ∩ (𝐴 × {1𝑜})) ≈ ω → ω ≼ 𝐴))
5743, 56jaod 394 . . . . . . 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 1901 . . . 4 (𝐴 ∈ V → (∃𝑥(ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴)) → ω ≼ 𝐴))
6116, 60syl5bi 232 . . 3 (𝐴 ∈ V → (ω ≼ (𝐴 +𝑐 𝐴) → ω ≼ 𝐴))
6214, 61mpcom 38 . 2 (ω ≼ (𝐴 +𝑐 𝐴) → ω ≼ 𝐴)
636, 62impbii 199 1 (ω ≼ 𝐴 ↔ ω ≼ (𝐴 +𝑐 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  Vcvv 3231  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210   class class class wbr 4685   × cxp 5141  dom cdm 5143  Oncon0 5761   Fn wfn 5921  (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-iun 4554  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-pred 5718  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-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-cda 9028
This theorem is referenced by:  infdif  9069
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