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Theorem cdainf 8966
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 7913 . . . . 5 Rel ≼
21brrelex2i 5124 . . . 4 (ω ≼ 𝐴𝐴 ∈ V)
3 cdadom3 8962 . . . 4 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → 𝐴 ≼ (𝐴 +𝑐 𝐴))
42, 2, 3syl2anc 692 . . 3 (ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 𝐴))
5 domtr 7961 . . 3 ((ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 𝐴)) → ω ≼ (𝐴 +𝑐 𝐴))
64, 5mpdan 701 . 2 (ω ≼ 𝐴 → ω ≼ (𝐴 +𝑐 𝐴))
7 infn0 8174 . . . 4 (ω ≼ (𝐴 +𝑐 𝐴) → (𝐴 +𝑐 𝐴) ≠ ∅)
8 cdafn 8943 . . . . . . . 8 +𝑐 Fn (V × V)
9 fndm 5953 . . . . . . . 8 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
108, 9ax-mp 5 . . . . . . 7 dom +𝑐 = (V × V)
1110ndmov 6778 . . . . . 6 (¬ (𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 +𝑐 𝐴) = ∅)
1211necon1ai 2817 . . . . 5 ((𝐴 +𝑐 𝐴) ≠ ∅ → (𝐴 ∈ V ∧ 𝐴 ∈ V))
1312simpld 475 . . . 4 ((𝐴 +𝑐 𝐴) ≠ ∅ → 𝐴 ∈ V)
147, 13syl 17 . . 3 (ω ≼ (𝐴 +𝑐 𝐴) → 𝐴 ∈ V)
15 ovex 6638 . . . . 5 (𝐴 +𝑐 𝐴) ∈ V
1615domen 7920 . . . 4 (ω ≼ (𝐴 +𝑐 𝐴) ↔ ∃𝑥(ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴)))
17 indi 3854 . . . . . . . . 9 (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) = ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜})))
18 simprr 795 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ⊆ (𝐴 +𝑐 𝐴))
19 simpl 473 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝐴 ∈ V)
20 cdaval 8944 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
2119, 19, 20syl2anc 692 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
2218, 21sseqtrd 3625 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ⊆ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
23 df-ss 3573 . . . . . . . . . 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 2669 . . . . . . . 8 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) = 𝑥)
26 ensym 7957 . . . . . . . . 9 (ω ≈ 𝑥𝑥 ≈ ω)
2726ad2antrl 763 . . . . . . . 8 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ≈ ω)
2825, 27eqbrtrd 4640 . . . . . . 7 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω)
2928ex 450 . . . . . 6 (𝐴 ∈ V → ((ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴)) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω))
30 cdainflem 8965 . . . . . . 7 (((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω → ((𝑥 ∩ (𝐴 × {∅})) ≈ ω ∨ (𝑥 ∩ (𝐴 × {1𝑜})) ≈ ω))
31 snex 4874 . . . . . . . . . . . 12 {∅} ∈ V
32 xpexg 6920 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
3331, 32mpan2 706 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 × {∅}) ∈ V)
34 inss2 3817 . . . . . . . . . . 11 (𝑥 ∩ (𝐴 × {∅})) ⊆ (𝐴 × {∅})
35 ssdomg 7953 . . . . . . . . . . 11 ((𝐴 × {∅}) ∈ V → ((𝑥 ∩ (𝐴 × {∅})) ⊆ (𝐴 × {∅}) → (𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅})))
3633, 34, 35mpisyl 21 . . . . . . . . . 10 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅}))
37 0ex 4755 . . . . . . . . . . 11 ∅ ∈ V
38 xpsneng 7997 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
3937, 38mpan2 706 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 × {∅}) ≈ 𝐴)
40 domentr 7967 . . . . . . . . . 10 (((𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅}) ∧ (𝐴 × {∅}) ≈ 𝐴) → (𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴)
4136, 39, 40syl2anc 692 . . . . . . . . 9 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴)
42 domen1 8054 . . . . . . . . 9 ((𝑥 ∩ (𝐴 × {∅})) ≈ ω → ((𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴 ↔ ω ≼ 𝐴))
4341, 42syl5ibcom 235 . . . . . . . 8 (𝐴 ∈ V → ((𝑥 ∩ (𝐴 × {∅})) ≈ ω → ω ≼ 𝐴))
44 snex 4874 . . . . . . . . . . . 12 {1𝑜} ∈ V
45 xpexg 6920 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ {1𝑜} ∈ V) → (𝐴 × {1𝑜}) ∈ V)
4644, 45mpan2 706 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 × {1𝑜}) ∈ V)
47 inss2 3817 . . . . . . . . . . 11 (𝑥 ∩ (𝐴 × {1𝑜})) ⊆ (𝐴 × {1𝑜})
48 ssdomg 7953 . . . . . . . . . . 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 7519 . . . . . . . . . . 11 1𝑜 ∈ On
51 xpsneng 7997 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
5250, 51mpan2 706 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 × {1𝑜}) ≈ 𝐴)
53 domentr 7967 . . . . . . . . . 10 (((𝑥 ∩ (𝐴 × {1𝑜})) ≼ (𝐴 × {1𝑜}) ∧ (𝐴 × {1𝑜}) ≈ 𝐴) → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴)
5449, 52, 53syl2anc 692 . . . . . . . . 9 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴)
55 domen1 8054 . . . . . . . . 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 395 . . . . . . 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 1858 . . . 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 383  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  Vcvv 3189  cun 3557  cin 3558  wss 3559  c0 3896  {csn 4153   class class class wbr 4618   × cxp 5077  dom cdm 5079  Oncon0 5687   Fn wfn 5847  (class class class)co 6610  ωcom 7019  1𝑜c1o 7505  cen 7904  cdom 7905   +𝑐 ccda 8941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-cda 8942
This theorem is referenced by:  infdif  8983
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