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Theorem infpssr 9168
Description: Dedekind infinity implies existence of a denumerable subset: take a single point witnessing the proper subset relation and iterate the embedding. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
infpssr ((𝑋𝐴𝑋𝐴) → ω ≼ 𝐴)

Proof of Theorem infpssr
Dummy variables 𝑦 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pssnel 4072 . . 3 (𝑋𝐴 → ∃𝑦(𝑦𝐴 ∧ ¬ 𝑦𝑋))
21adantr 480 . 2 ((𝑋𝐴𝑋𝐴) → ∃𝑦(𝑦𝐴 ∧ ¬ 𝑦𝑋))
3 eldif 3617 . . . 4 (𝑦 ∈ (𝐴𝑋) ↔ (𝑦𝐴 ∧ ¬ 𝑦𝑋))
4 pssss 3735 . . . . . 6 (𝑋𝐴𝑋𝐴)
5 bren 8006 . . . . . . . 8 (𝑋𝐴 ↔ ∃𝑓 𝑓:𝑋1-1-onto𝐴)
6 simpr 476 . . . . . . . . . . . . 13 (((𝑦 ∈ (𝐴𝑋) ∧ 𝑋𝐴) ∧ 𝑓:𝑋1-1-onto𝐴) → 𝑓:𝑋1-1-onto𝐴)
7 f1ofo 6182 . . . . . . . . . . . . 13 (𝑓:𝑋1-1-onto𝐴𝑓:𝑋onto𝐴)
8 forn 6156 . . . . . . . . . . . . 13 (𝑓:𝑋onto𝐴 → ran 𝑓 = 𝐴)
96, 7, 83syl 18 . . . . . . . . . . . 12 (((𝑦 ∈ (𝐴𝑋) ∧ 𝑋𝐴) ∧ 𝑓:𝑋1-1-onto𝐴) → ran 𝑓 = 𝐴)
10 vex 3234 . . . . . . . . . . . . 13 𝑓 ∈ V
1110rnex 7142 . . . . . . . . . . . 12 ran 𝑓 ∈ V
129, 11syl6eqelr 2739 . . . . . . . . . . 11 (((𝑦 ∈ (𝐴𝑋) ∧ 𝑋𝐴) ∧ 𝑓:𝑋1-1-onto𝐴) → 𝐴 ∈ V)
13 simplr 807 . . . . . . . . . . . 12 (((𝑦 ∈ (𝐴𝑋) ∧ 𝑋𝐴) ∧ 𝑓:𝑋1-1-onto𝐴) → 𝑋𝐴)
14 simpll 805 . . . . . . . . . . . 12 (((𝑦 ∈ (𝐴𝑋) ∧ 𝑋𝐴) ∧ 𝑓:𝑋1-1-onto𝐴) → 𝑦 ∈ (𝐴𝑋))
15 eqid 2651 . . . . . . . . . . . 12 (rec(𝑓, 𝑦) ↾ ω) = (rec(𝑓, 𝑦) ↾ ω)
1613, 6, 14, 15infpssrlem5 9167 . . . . . . . . . . 11 (((𝑦 ∈ (𝐴𝑋) ∧ 𝑋𝐴) ∧ 𝑓:𝑋1-1-onto𝐴) → (𝐴 ∈ V → ω ≼ 𝐴))
1712, 16mpd 15 . . . . . . . . . 10 (((𝑦 ∈ (𝐴𝑋) ∧ 𝑋𝐴) ∧ 𝑓:𝑋1-1-onto𝐴) → ω ≼ 𝐴)
1817ex 449 . . . . . . . . 9 ((𝑦 ∈ (𝐴𝑋) ∧ 𝑋𝐴) → (𝑓:𝑋1-1-onto𝐴 → ω ≼ 𝐴))
1918exlimdv 1901 . . . . . . . 8 ((𝑦 ∈ (𝐴𝑋) ∧ 𝑋𝐴) → (∃𝑓 𝑓:𝑋1-1-onto𝐴 → ω ≼ 𝐴))
205, 19syl5bi 232 . . . . . . 7 ((𝑦 ∈ (𝐴𝑋) ∧ 𝑋𝐴) → (𝑋𝐴 → ω ≼ 𝐴))
2120ex 449 . . . . . 6 (𝑦 ∈ (𝐴𝑋) → (𝑋𝐴 → (𝑋𝐴 → ω ≼ 𝐴)))
224, 21syl5 34 . . . . 5 (𝑦 ∈ (𝐴𝑋) → (𝑋𝐴 → (𝑋𝐴 → ω ≼ 𝐴)))
2322impd 446 . . . 4 (𝑦 ∈ (𝐴𝑋) → ((𝑋𝐴𝑋𝐴) → ω ≼ 𝐴))
243, 23sylbir 225 . . 3 ((𝑦𝐴 ∧ ¬ 𝑦𝑋) → ((𝑋𝐴𝑋𝐴) → ω ≼ 𝐴))
2524exlimiv 1898 . 2 (∃𝑦(𝑦𝐴 ∧ ¬ 𝑦𝑋) → ((𝑋𝐴𝑋𝐴) → ω ≼ 𝐴))
262, 25mpcom 38 1 ((𝑋𝐴𝑋𝐴) → ω ≼ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  cdif 3604  wss 3607  wpss 3608   class class class wbr 4685  ccnv 5142  ran crn 5144  cres 5145  ontowfo 5924  1-1-ontowf1o 5925  ωcom 7107  reccrdg 7550  cen 7994  cdom 7995
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-rep 4804  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-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-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-en 7998  df-dom 7999
This theorem is referenced by:  isfin4-2  9174
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