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Theorem infn0 9341
Description: An infinite set is not empty. For a shorter proof using ax-un 7746, see infn0ALT 9342. (Contributed by NM, 23-Oct-2004.) Avoid ax-un 7746. (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 8989 . 2 (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1𝐴)
2 peano1 7900 . . . . . 6 ∅ ∈ ω
3 f1f1orn 6854 . . . . . . . . 9 (𝑓:ω–1-1𝐴𝑓:ω–1-1-onto→ran 𝑓)
43adantr 479 . . . . . . . 8 ((𝑓:ω–1-1𝐴𝐴 = ∅) → 𝑓:ω–1-1-onto→ran 𝑓)
5 f1f 6798 . . . . . . . . . . 11 (𝑓:ω–1-1𝐴𝑓:ω⟶𝐴)
65frnd 6736 . . . . . . . . . 10 (𝑓:ω–1-1𝐴 → ran 𝑓𝐴)
7 sseq0 4404 . . . . . . . . . 10 ((ran 𝑓𝐴𝐴 = ∅) → ran 𝑓 = ∅)
86, 7sylan 578 . . . . . . . . 9 ((𝑓:ω–1-1𝐴𝐴 = ∅) → ran 𝑓 = ∅)
98f1oeq3d 6840 . . . . . . . 8 ((𝑓:ω–1-1𝐴𝐴 = ∅) → (𝑓:ω–1-1-onto→ran 𝑓𝑓:ω–1-1-onto→∅))
104, 9mpbid 231 . . . . . . 7 ((𝑓:ω–1-1𝐴𝐴 = ∅) → 𝑓:ω–1-1-onto→∅)
11 f1ocnv 6855 . . . . . . 7 (𝑓:ω–1-1-onto→∅ → 𝑓:∅–1-1-onto→ω)
12 noel 4333 . . . . . . . 8 ¬ ∅ ∈ ∅
13 f1o00 6878 . . . . . . . . . 10 (𝑓:∅–1-1-onto→ω ↔ (𝑓 = ∅ ∧ ω = ∅))
1413simprbi 495 . . . . . . . . 9 (𝑓:∅–1-1-onto→ω → ω = ∅)
1514eleq2d 2812 . . . . . . . 8 (𝑓:∅–1-1-onto→ω → (∅ ∈ ω ↔ ∅ ∈ ∅))
1612, 15mtbiri 326 . . . . . . 7 (𝑓:∅–1-1-onto→ω → ¬ ∅ ∈ ω)
1710, 11, 163syl 18 . . . . . 6 ((𝑓:ω–1-1𝐴𝐴 = ∅) → ¬ ∅ ∈ ω)
182, 17mt2 199 . . . . 5 ¬ (𝑓:ω–1-1𝐴𝐴 = ∅)
1918imnani 399 . . . 4 (𝑓:ω–1-1𝐴 → ¬ 𝐴 = ∅)
2019neqned 2937 . . 3 (𝑓:ω–1-1𝐴𝐴 ≠ ∅)
2120exlimiv 1926 . 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 394   = wceq 1534  wex 1774  wcel 2099  wne 2930  wss 3947  c0 4325   class class class wbr 5153  ccnv 5681  ran crn 5683  1-1wf1 6551  1-1-ontowf1o 6553  ωcom 7876  cdom 8972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-mo 2529  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-ord 6379  df-on 6380  df-lim 6381  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-om 7877  df-dom 8976
This theorem is referenced by:  infpwfien  10105  infxp  10258  infpss  10260  alephmul  10621  csdfil  23889
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