Theorem infn0 9212

Description: An infinite set is not empty. For a shorter proof using ax-un 7689, see infn0ALT 9213. (Contributed by NM, 23-Oct-2004.) Avoid ax-un 7689. (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 8906	. 2 ⊢ (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1→𝐴)
2		peano1 7840	. . . . . 6 ⊢ ∅ ∈ ω
3		f1f1orn 6791	. . . . . . . . 9 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓:ω–1-1-onto→ran 𝑓)
4	3	adantr 480	. . . . . . . 8 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝐴 = ∅) → 𝑓:ω–1-1-onto→ran 𝑓)
5		f1f 6736	. . . . . . . . . . 11 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓:ω⟶𝐴)
6	5	frnd 6676	. . . . . . . . . 10 ⊢ (𝑓:ω–1-1→𝐴 → ran 𝑓 ⊆ 𝐴)
7		sseq0 4343	. . . . . . . . . 10 ⊢ ((ran 𝑓 ⊆ 𝐴 ∧ 𝐴 = ∅) → ran 𝑓 = ∅)
8	6, 7	sylan 581	. . . . . . . . 9 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝐴 = ∅) → ran 𝑓 = ∅)
9	8	f1oeq3d 6777	. . . . . . . 8 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝐴 = ∅) → (𝑓:ω–1-1-onto→ran 𝑓 ↔ 𝑓:ω–1-1-onto→∅))
10	4, 9	mpbid 232	. . . . . . 7 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝐴 = ∅) → 𝑓:ω–1-1-onto→∅)
11		f1ocnv 6792	. . . . . . 7 ⊢ (𝑓:ω–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→ω)
12		noel 4278	. . . . . . . 8 ⊢ ¬ ∅ ∈ ∅
13		f1o00 6815	. . . . . . . . . 10 ⊢ (◡𝑓:∅–1-1-onto→ω ↔ (◡𝑓 = ∅ ∧ ω = ∅))
14	13	simprbi 497	. . . . . . . . 9 ⊢ (◡𝑓:∅–1-1-onto→ω → ω = ∅)
15	14	eleq2d 2822	. . . . . . . 8 ⊢ (◡𝑓:∅–1-1-onto→ω → (∅ ∈ ω ↔ ∅ ∈ ∅))
16	12, 15	mtbiri 327	. . . . . . 7 ⊢ (◡𝑓:∅–1-1-onto→ω → ¬ ∅ ∈ ω)
17	10, 11, 16	3syl 18	. . . . . 6 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝐴 = ∅) → ¬ ∅ ∈ ω)
18	2, 17	mt2 200	. . . . 5 ⊢ ¬ (𝑓:ω–1-1→𝐴 ∧ 𝐴 = ∅)
19	18	imnani 400	. . . 4 ⊢ (𝑓:ω–1-1→𝐴 → ¬ 𝐴 = ∅)
20	19	neqned 2939	. . 3 ⊢ (𝑓:ω–1-1→𝐴 → 𝐴 ≠ ∅)
21	20	exlimiv 1932	. 2 ⊢ (∃𝑓 𝑓:ω–1-1→𝐴 → 𝐴 ≠ ∅)
22	1, 21	syl 17	1 ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932   ⊆ wss 3889  ∅c0 4273   class class class wbr 5085  ^◡ccnv 5630  ran crn 5632  –1-1→wf1 6495  –1-1-onto→wf1o 6497  ωcom 7817   ≼ cdom 8891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-ord 6326  df-on 6327  df-lim 6328  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-om 7818  df-dom 8895

This theorem is referenced by:  infpwfien  9984  infxp  10136  infpss  10138  alephmul  10501  csdfil  23859