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Theorem infn0 9341
Description: An infinite set is not empty. For a shorter proof using ax-un 7756, see infn0ALT 9342. (Contributed by NM, 23-Oct-2004.) Avoid ax-un 7756. (Revised by BTernaryTau, 8-Jan-2025.)
Assertion
Ref Expression
infn0 (ω ≼ 𝐴𝐴 ≠ ∅)

Proof of Theorem infn0
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 brdomi 9000 . 2 (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1𝐴)
2 peano1 7911 . . . . . 6 ∅ ∈ ω
3 f1f1orn 6858 . . . . . . . . 9 (𝑓:ω–1-1𝐴𝑓:ω–1-1-onto→ran 𝑓)
43adantr 480 . . . . . . . 8 ((𝑓:ω–1-1𝐴𝐴 = ∅) → 𝑓:ω–1-1-onto→ran 𝑓)
5 f1f 6803 . . . . . . . . . . 11 (𝑓:ω–1-1𝐴𝑓:ω⟶𝐴)
65frnd 6743 . . . . . . . . . 10 (𝑓:ω–1-1𝐴 → ran 𝑓𝐴)
7 sseq0 4402 . . . . . . . . . 10 ((ran 𝑓𝐴𝐴 = ∅) → ran 𝑓 = ∅)
86, 7sylan 580 . . . . . . . . 9 ((𝑓:ω–1-1𝐴𝐴 = ∅) → ran 𝑓 = ∅)
98f1oeq3d 6844 . . . . . . . 8 ((𝑓:ω–1-1𝐴𝐴 = ∅) → (𝑓:ω–1-1-onto→ran 𝑓𝑓:ω–1-1-onto→∅))
104, 9mpbid 232 . . . . . . 7 ((𝑓:ω–1-1𝐴𝐴 = ∅) → 𝑓:ω–1-1-onto→∅)
11 f1ocnv 6859 . . . . . . 7 (𝑓:ω–1-1-onto→∅ → 𝑓:∅–1-1-onto→ω)
12 noel 4337 . . . . . . . 8 ¬ ∅ ∈ ∅
13 f1o00 6882 . . . . . . . . . 10 (𝑓:∅–1-1-onto→ω ↔ (𝑓 = ∅ ∧ ω = ∅))
1413simprbi 496 . . . . . . . . 9 (𝑓:∅–1-1-onto→ω → ω = ∅)
1514eleq2d 2826 . . . . . . . 8 (𝑓:∅–1-1-onto→ω → (∅ ∈ ω ↔ ∅ ∈ ∅))
1612, 15mtbiri 327 . . . . . . 7 (𝑓:∅–1-1-onto→ω → ¬ ∅ ∈ ω)
1710, 11, 163syl 18 . . . . . 6 ((𝑓:ω–1-1𝐴𝐴 = ∅) → ¬ ∅ ∈ ω)
182, 17mt2 200 . . . . 5 ¬ (𝑓:ω–1-1𝐴𝐴 = ∅)
1918imnani 400 . . . 4 (𝑓:ω–1-1𝐴 → ¬ 𝐴 = ∅)
2019neqned 2946 . . 3 (𝑓:ω–1-1𝐴𝐴 ≠ ∅)
2120exlimiv 1929 . 2 (∃𝑓 𝑓:ω–1-1𝐴𝐴 ≠ ∅)
221, 21syl 17 1 (ω ≼ 𝐴𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1539  wex 1778  wcel 2107  wne 2939  wss 3950  c0 4332   class class class wbr 5142  ccnv 5683  ran crn 5685  1-1wf1 6557  1-1-ontowf1o 6559  ωcom 7888  cdom 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-ord 6386  df-on 6387  df-lim 6388  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-om 7889  df-dom 8988
This theorem is referenced by:  infpwfien  10103  infxp  10255  infpss  10257  alephmul  10619  csdfil  23903
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