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Theorem cdainf 9214
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 8113 . . . . 5 Rel ≼
21brrelex2i 5297 . . . 4 (ω ≼ 𝐴𝐴 ∈ V)
3 cdadom3 9210 . . . 4 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → 𝐴 ≼ (𝐴 +𝑐 𝐴))
42, 2, 3syl2anc 573 . . 3 (ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 𝐴))
5 domtr 8160 . . 3 ((ω ≼ 𝐴𝐴 ≼ (𝐴 +𝑐 𝐴)) → ω ≼ (𝐴 +𝑐 𝐴))
64, 5mpdan 667 . 2 (ω ≼ 𝐴 → ω ≼ (𝐴 +𝑐 𝐴))
7 infn0 8376 . . . 4 (ω ≼ (𝐴 +𝑐 𝐴) → (𝐴 +𝑐 𝐴) ≠ ∅)
8 cdafn 9191 . . . . . . . 8 +𝑐 Fn (V × V)
9 fndm 6128 . . . . . . . 8 ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V))
108, 9ax-mp 5 . . . . . . 7 dom +𝑐 = (V × V)
1110ndmov 6963 . . . . . 6 (¬ (𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 +𝑐 𝐴) = ∅)
1211necon1ai 2970 . . . . 5 ((𝐴 +𝑐 𝐴) ≠ ∅ → (𝐴 ∈ V ∧ 𝐴 ∈ V))
1312simpld 482 . . . 4 ((𝐴 +𝑐 𝐴) ≠ ∅ → 𝐴 ∈ V)
147, 13syl 17 . . 3 (ω ≼ (𝐴 +𝑐 𝐴) → 𝐴 ∈ V)
15 ovex 6821 . . . . 5 (𝐴 +𝑐 𝐴) ∈ V
1615domen 8120 . . . 4 (ω ≼ (𝐴 +𝑐 𝐴) ↔ ∃𝑥(ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴)))
17 indi 4022 . . . . . . . . 9 (𝑥 ∩ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜}))) = ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜})))
18 simprr 756 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ⊆ (𝐴 +𝑐 𝐴))
19 simpl 468 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝐴 ∈ V)
20 cdaval 9192 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
2119, 19, 20syl2anc 573 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → (𝐴 +𝑐 𝐴) = ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
2218, 21sseqtrd 3790 . . . . . . . . . 10 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ⊆ ((𝐴 × {∅}) ∪ (𝐴 × {1𝑜})))
23 df-ss 3737 . . . . . . . . . 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 2819 . . . . . . . 8 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) = 𝑥)
26 ensym 8156 . . . . . . . . 9 (ω ≈ 𝑥𝑥 ≈ ω)
2726ad2antrl 707 . . . . . . . 8 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → 𝑥 ≈ ω)
2825, 27eqbrtrd 4808 . . . . . . 7 ((𝐴 ∈ V ∧ (ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴))) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω)
2928ex 397 . . . . . 6 (𝐴 ∈ V → ((ω ≈ 𝑥𝑥 ⊆ (𝐴 +𝑐 𝐴)) → ((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω))
30 cdainflem 9213 . . . . . . 7 (((𝑥 ∩ (𝐴 × {∅})) ∪ (𝑥 ∩ (𝐴 × {1𝑜}))) ≈ ω → ((𝑥 ∩ (𝐴 × {∅})) ≈ ω ∨ (𝑥 ∩ (𝐴 × {1𝑜})) ≈ ω))
31 snex 5036 . . . . . . . . . . . 12 {∅} ∈ V
32 xpexg 7105 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
3331, 32mpan2 671 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 × {∅}) ∈ V)
34 inss2 3982 . . . . . . . . . . 11 (𝑥 ∩ (𝐴 × {∅})) ⊆ (𝐴 × {∅})
35 ssdomg 8153 . . . . . . . . . . 11 ((𝐴 × {∅}) ∈ V → ((𝑥 ∩ (𝐴 × {∅})) ⊆ (𝐴 × {∅}) → (𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅})))
3633, 34, 35mpisyl 21 . . . . . . . . . 10 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅}))
37 0ex 4924 . . . . . . . . . . 11 ∅ ∈ V
38 xpsneng 8199 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴)
3937, 38mpan2 671 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 × {∅}) ≈ 𝐴)
40 domentr 8166 . . . . . . . . . 10 (((𝑥 ∩ (𝐴 × {∅})) ≼ (𝐴 × {∅}) ∧ (𝐴 × {∅}) ≈ 𝐴) → (𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴)
4136, 39, 40syl2anc 573 . . . . . . . . 9 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴)
42 domen1 8256 . . . . . . . . 9 ((𝑥 ∩ (𝐴 × {∅})) ≈ ω → ((𝑥 ∩ (𝐴 × {∅})) ≼ 𝐴 ↔ ω ≼ 𝐴))
4341, 42syl5ibcom 235 . . . . . . . 8 (𝐴 ∈ V → ((𝑥 ∩ (𝐴 × {∅})) ≈ ω → ω ≼ 𝐴))
44 snex 5036 . . . . . . . . . . . 12 {1𝑜} ∈ V
45 xpexg 7105 . . . . . . . . . . . 12 ((𝐴 ∈ V ∧ {1𝑜} ∈ V) → (𝐴 × {1𝑜}) ∈ V)
4644, 45mpan2 671 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 × {1𝑜}) ∈ V)
47 inss2 3982 . . . . . . . . . . 11 (𝑥 ∩ (𝐴 × {1𝑜})) ⊆ (𝐴 × {1𝑜})
48 ssdomg 8153 . . . . . . . . . . 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 7718 . . . . . . . . . . 11 1𝑜 ∈ On
51 xpsneng 8199 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴)
5250, 51mpan2 671 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 × {1𝑜}) ≈ 𝐴)
53 domentr 8166 . . . . . . . . . 10 (((𝑥 ∩ (𝐴 × {1𝑜})) ≼ (𝐴 × {1𝑜}) ∧ (𝐴 × {1𝑜}) ≈ 𝐴) → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴)
5449, 52, 53syl2anc 573 . . . . . . . . 9 (𝐴 ∈ V → (𝑥 ∩ (𝐴 × {1𝑜})) ≼ 𝐴)
55 domen1 8256 . . . . . . . . 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 848 . . . . . . 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 2013 . . . 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  wa 382  wo 836   = wceq 1631  wex 1852  wcel 2145  wne 2943  Vcvv 3351  cun 3721  cin 3722  wss 3723  c0 4063  {csn 4316   class class class wbr 4786   × cxp 5247  dom cdm 5249  Oncon0 5864   Fn wfn 6024  (class class class)co 6791  ωcom 7210  1𝑜c1o 7704  cen 8104  cdom 8105   +𝑐 ccda 9189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-oadd 7715  df-er 7894  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-cda 9190
This theorem is referenced by:  infdif  9231
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