Theorem infn0 9325

Description: An infinite set is not empty. For a shorter proof using ax-un 7778, see infn0ALT 9326. (Contributed by NM, 23-Oct-2004.) Avoid ax-un 7778. (Revised by BTernaryTau, 8-Jan-2025.)

Assertion

Ref	Expression
infn0	⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅)

Proof of Theorem infn0

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 9017	. 2 ⊢ (ω ≼ 𝐴 → ∃𝑓 𝑓:ω⟶1-1𝐴)
2		peano1 7927	. . . . . 6 ⊢ ∅ ∈ ω
3		f1f1orn 6861	. . . . . . . . 9 ⊢ (𝑓:ω⟶1-1𝐴 → 𝑓:ω⟶1-1ontoran 𝑓)
4	3	adantr 486	. . . . . . . 8 ⊢ ((𝑓:ω⟶1-1𝐴 ∧ 𝐴 = ∅) → 𝑓:ω⟶1-1ontoran 𝑓)
5		f1f 6803	. . . . . . . . . . 11 ⊢ (𝑓:ω⟶1-1𝐴 → 𝑓:ω⟶𝐴)
6	5	frnd 6743	. . . . . . . . . 10 ⊢ (𝑓:ω⟶1-1𝐴 → ran 𝑓 ⊆ 𝐴)
7		sseq0 4357	. . . . . . . . . 10 ⊢ ((ran 𝑓 ⊆ 𝐴 ∧ 𝐴 = ∅) → ran 𝑓 = ∅)
8	6, 7	sylan 592	. . . . . . . . 9 ⊢ ((𝑓:ω⟶1-1𝐴 ∧ 𝐴 = ∅) → ran 𝑓 = ∅)
9	8	f1oeq3d 6846	. . . . . . . 8 ⊢ ((𝑓:ω⟶1-1𝐴 ∧ 𝐴 = ∅) → (𝑓:ω⟶1-1ontoran 𝑓 ↔ 𝑓:ω⟶1-1onto∅))
10	4, 9	mpbid 235	. . . . . . 7 ⊢ ((𝑓:ω⟶1-1𝐴 ∧ 𝐴 = ∅) → 𝑓:ω⟶1-1onto∅)
11		f1ocnv 6862	. . . . . . 7 ⊢ (𝑓:ω⟶1-1onto∅ → ◡𝑓:∅⟶1-1ontoω)
12		noel 4286	. . . . . . . 8 ⊢ ¬ ∅ ∈ ∅
13		f1o00 6886	. . . . . . . . . 10 ⊢ (◡𝑓:∅⟶1-1ontoω ↔ (◡𝑓 = ∅ ∧ ω = ∅))
14	13	simprbi 503	. . . . . . . . 9 ⊢ (◡𝑓:∅⟶1-1ontoω → ω = ∅)
15	14	eleq2d 2850	. . . . . . . 8 ⊢ (◡𝑓:∅⟶1-1ontoω → (∅ ∈ ω ↔ ∅ ∈ ∅))
16	12, 15	mtbiri 330	. . . . . . 7 ⊢ (◡𝑓:∅⟶1-1ontoω → ¬ ∅ ∈ ω)
17	10, 11, 16	3syl 19	. . . . . 6 ⊢ ((𝑓:ω⟶1-1𝐴 ∧ 𝐴 = ∅) → ¬ ∅ ∈ ω)
18	2, 17	mt2 203	. . . . 5 ⊢ ¬ (𝑓:ω⟶1-1𝐴 ∧ 𝐴 = ∅)
19	18	imnani 406	. . . 4 ⊢ (𝑓:ω⟶1-1𝐴 → ¬ 𝐴 = ∅)
20	19	neqned 2966	. . 3 ⊢ (𝑓:ω⟶1-1𝐴 → 𝐴 ≠ ∅)
21	20	exlimiv 1965	. 2 ⊢ (∃𝑓 𝑓:ω⟶1-1𝐴 → 𝐴 ≠ ∅)
22	1, 21	syl 18	1 ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959   ⊆ wss 3900  ∅c0 4281   class class class wbr 5106  ^◡ccnv 5655  ran crn 5657  ⟶1-1wf1 6558  ⟶1-1ontowf1o 6560  ωcom 7904   ≼ cdom 9002

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-ord 6386  df-on 6387  df-lim 6388  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-om 7905  df-dom 9006

This theorem is used by:  infpwfien  10171  infxp  10322  infpss  10324  alephmul  10693  csdfil  24246  soinfdom  35813