MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cdainf Structured version   Visualization version   GIF version

Theorem cdainf 8897
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 7847 . . . . 5 Rel ≼
21brrelex2i 5083 . . . 4 (ω ≼ 𝐴𝐴 ∈ V)
3 cdadom3 8893 . . . 4 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → 𝐴 ≼ (𝐴 +𝑐 𝐴))
42, 2, 3syl2anc 691 . . 3 (ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 𝐴))
5 domtr 7895 . . 3 ((ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 𝐴)) → ω ≼ (𝐴 +𝑐 𝐴))
64, 5mpdan 699 . 2 (ω ≼ 𝐴 → ω ≼ (𝐴 +𝑐 𝐴))
7 infn0 8107 . . . 4 (ω ≼ (𝐴 +𝑐 𝐴) → (𝐴 +𝑐 𝐴) ≠ ∅)
8 cdafn 8874 . . . . . . . 8 +𝑐 Fn (V × V)
9 fndm 5904 . . . . . . . 8 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
108, 9ax-mp 5 . . . . . . 7 dom +𝑐 = (V × V)
1110ndmov 6716 . . . . . 6 (¬ (𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 +𝑐 𝐴) = ∅)
1211necon1ai 2809 . . . . 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 6577 . . . . 5 (𝐴 +𝑐 𝐴) ∈ V
1615domen 7854 . . . 4 (ω ≼ (𝐴 +𝑐 𝐴) ↔ ∃𝑥(ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴)))
17 indi 3832 . . . . . . . . 9 (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) = ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜})))
18 simprr 792 . . . . . . . . . . 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 8875 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
2119, 19, 20syl2anc 691 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
2218, 21sseqtrd 3604 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ⊆ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
23 df-ss 3554 . . . . . . . . . 10 (𝑥 ⊆ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})) ↔ (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) = 𝑥)
2422, 23sylib 207 . . . . . . . . 9 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) = 𝑥)
2517, 24syl5eqr 2658 . . . . . . . 8 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) = 𝑥)
26 ensym 7891 . . . . . . . . 9 (ω ≈ 𝑥𝑥 ≈ ω)
2726ad2antrl 760 . . . . . . . 8 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ≈ ω)
2825, 27eqbrtrd 4605 . . . . . . 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 8896 . . . . . . 7 (((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω → ((𝑥 ∩ (𝐴 × {∅})) ≈ ω ∨ (𝑥 ∩ (𝐴 × {1𝑜})) ≈ ω))
31 snex 4835 . . . . . . . . . . . 12 {∅} ∈ V
32 xpexg 6858 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
3331, 32mpan2 703 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 × {∅}) ∈ V)
34 inss2 3796 . . . . . . . . . . 11 (𝑥 ∩ (𝐴 × {∅})) ⊆ (𝐴 × {∅})
35 ssdomg 7887 . . . . . . . . . . 11 ((𝐴 × {∅}) ∈ V → ((𝑥 ∩ (𝐴 × {∅})) ⊆ (𝐴 × {∅}) → (𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅})))
3633, 34, 35mpisyl 21 . . . . . . . . . 10 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅}))
37 0ex 4718 . . . . . . . . . . 11 ∅ ∈ V
38 xpsneng 7930 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
3937, 38mpan2 703 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 × {∅}) ≈ 𝐴)
40 domentr 7901 . . . . . . . . . 10 (((𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅}) ∧ (𝐴 × {∅}) ≈ 𝐴) → (𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴)
4136, 39, 40syl2anc 691 . . . . . . . . 9 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴)
42 domen1 7987 . . . . . . . . 9 ((𝑥 ∩ (𝐴 × {∅})) ≈ ω → ((𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴 ↔ ω ≼ 𝐴))
4341, 42syl5ibcom 234 . . . . . . . 8 (𝐴 ∈ V → ((𝑥 ∩ (𝐴 × {∅})) ≈ ω → ω ≼ 𝐴))
44 snex 4835 . . . . . . . . . . . 12 {1𝑜} ∈ V
45 xpexg 6858 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ {1𝑜} ∈ V) → (𝐴 × {1𝑜}) ∈ V)
4644, 45mpan2 703 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 × {1𝑜}) ∈ V)
47 inss2 3796 . . . . . . . . . . 11 (𝑥 ∩ (𝐴 × {1𝑜})) ⊆ (𝐴 × {1𝑜})
48 ssdomg 7887 . . . . . . . . . . 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 7454 . . . . . . . . . . 11 1𝑜 ∈ On
51 xpsneng 7930 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
5250, 51mpan2 703 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 × {1𝑜}) ≈ 𝐴)
53 domentr 7901 . . . . . . . . . 10 (((𝑥 ∩ (𝐴 × {1𝑜})) ≼ (𝐴 × {1𝑜}) ∧ (𝐴 × {1𝑜}) ≈ 𝐴) → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴)
5449, 52, 53syl2anc 691 . . . . . . . . 9 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴)
55 domen1 7987 . . . . . . . . 9 ((𝑥 ∩ (𝐴 × {1𝑜})) ≈ ω → ((𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴 ↔ ω ≼ 𝐴))
5654, 55syl5ibcom 234 . . . . . . . 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 33 . . . . . 6 (𝐴 ∈ V → (((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω → ω ≼ 𝐴))
5929, 58syld 46 . . . . 5 (𝐴 ∈ V → ((ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴)) → ω ≼ 𝐴))
6059exlimdv 1848 . . . 4 (𝐴 ∈ V → (∃𝑥(ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴)) → ω ≼ 𝐴))
6116, 60syl5bi 231 . . 3 (𝐴 ∈ V → (ω ≼ (𝐴 +𝑐 𝐴) → ω ≼ 𝐴))
6214, 61mpcom 37 . 2 (ω ≼ (𝐴 +𝑐 𝐴) → ω ≼ 𝐴)
636, 62impbii 198 1 (ω ≼ 𝐴 ↔ ω ≼ (𝐴 +𝑐 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wo 382  wa 383   = wceq 1475  wex 1695  wcel 1977  wne 2780  Vcvv 3173  cun 3538  cin 3539  wss 3540  c0 3874  {csn 4125   class class class wbr 4583   × cxp 5036  dom cdm 5038  Oncon0 5640   Fn wfn 5799  (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-iun 4457  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-pred 5597  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-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-cda 8873
This theorem is referenced by:  infdif  8914
  Copyright terms: Public domain W3C validator