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Theorem isinfinf 6759
Description: An infinite set contains subsets of arbitrarily large finite cardinality. (Contributed by Jim Kingdon, 15-Jun-2022.)
Assertion
Ref Expression
isinfinf (ω ≼ 𝐴 → ∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛))
Distinct variable group:   𝐴,𝑛,𝑥

Proof of Theorem isinfinf
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 brdomi 6611 . . . 4 (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1𝐴)
21adantr 274 . . 3 ((ω ≼ 𝐴𝑛 ∈ ω) → ∃𝑓 𝑓:ω–1-1𝐴)
3 vex 2663 . . . . 5 𝑓 ∈ V
4 imaexg 4863 . . . . 5 (𝑓 ∈ V → (𝑓𝑛) ∈ V)
53, 4ax-mp 5 . . . 4 (𝑓𝑛) ∈ V
6 imassrn 4862 . . . . . 6 (𝑓𝑛) ⊆ ran 𝑓
7 simpr 109 . . . . . . 7 (((ω ≼ 𝐴𝑛 ∈ ω) ∧ 𝑓:ω–1-1𝐴) → 𝑓:ω–1-1𝐴)
8 f1f 5298 . . . . . . 7 (𝑓:ω–1-1𝐴𝑓:ω⟶𝐴)
9 frn 5251 . . . . . . 7 (𝑓:ω⟶𝐴 → ran 𝑓𝐴)
107, 8, 93syl 17 . . . . . 6 (((ω ≼ 𝐴𝑛 ∈ ω) ∧ 𝑓:ω–1-1𝐴) → ran 𝑓𝐴)
116, 10sstrid 3078 . . . . 5 (((ω ≼ 𝐴𝑛 ∈ ω) ∧ 𝑓:ω–1-1𝐴) → (𝑓𝑛) ⊆ 𝐴)
12 ordom 4490 . . . . . . . 8 Ord ω
13 ordelss 4271 . . . . . . . 8 ((Ord ω ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
1412, 13mpan 420 . . . . . . 7 (𝑛 ∈ ω → 𝑛 ⊆ ω)
1514ad2antlr 480 . . . . . 6 (((ω ≼ 𝐴𝑛 ∈ ω) ∧ 𝑓:ω–1-1𝐴) → 𝑛 ⊆ ω)
16 simplr 504 . . . . . 6 (((ω ≼ 𝐴𝑛 ∈ ω) ∧ 𝑓:ω–1-1𝐴) → 𝑛 ∈ ω)
17 f1imaeng 6654 . . . . . 6 ((𝑓:ω–1-1𝐴𝑛 ⊆ ω ∧ 𝑛 ∈ ω) → (𝑓𝑛) ≈ 𝑛)
187, 15, 16, 17syl3anc 1201 . . . . 5 (((ω ≼ 𝐴𝑛 ∈ ω) ∧ 𝑓:ω–1-1𝐴) → (𝑓𝑛) ≈ 𝑛)
1911, 18jca 304 . . . 4 (((ω ≼ 𝐴𝑛 ∈ ω) ∧ 𝑓:ω–1-1𝐴) → ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛))
20 sseq1 3090 . . . . . 6 (𝑥 = (𝑓𝑛) → (𝑥𝐴 ↔ (𝑓𝑛) ⊆ 𝐴))
21 breq1 3902 . . . . . 6 (𝑥 = (𝑓𝑛) → (𝑥𝑛 ↔ (𝑓𝑛) ≈ 𝑛))
2220, 21anbi12d 464 . . . . 5 (𝑥 = (𝑓𝑛) → ((𝑥𝐴𝑥𝑛) ↔ ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)))
2322spcegv 2748 . . . 4 ((𝑓𝑛) ∈ V → (((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → ∃𝑥(𝑥𝐴𝑥𝑛)))
245, 19, 23mpsyl 65 . . 3 (((ω ≼ 𝐴𝑛 ∈ ω) ∧ 𝑓:ω–1-1𝐴) → ∃𝑥(𝑥𝐴𝑥𝑛))
252, 24exlimddv 1854 . 2 ((ω ≼ 𝐴𝑛 ∈ ω) → ∃𝑥(𝑥𝐴𝑥𝑛))
2625ralrimiva 2482 1 (ω ≼ 𝐴 → ∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wex 1453  wcel 1465  wral 2393  Vcvv 2660  wss 3041   class class class wbr 3899  Ord word 4254  ωcom 4474  ran crn 4510  cima 4512  wf 5089  1-1wf1 5090  cen 6600  cdom 6601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-er 6397  df-en 6603  df-dom 6604
This theorem is referenced by: (None)
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