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| 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.) | 
| Ref | Expression | 
|---|---|
| infn0 | ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | brdomi 9000 | . 2 ⊢ (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1→𝐴) | |
| 2 | peano1 7911 | . . . . . 6 ⊢ ∅ ∈ ω | |
| 3 | f1f1orn 6858 | . . . . . . . . 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 6803 | . . . . . . . . . . 11 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓:ω⟶𝐴) | |
| 6 | 5 | frnd 6743 | . . . . . . . . . 10 ⊢ (𝑓:ω–1-1→𝐴 → ran 𝑓 ⊆ 𝐴) | 
| 7 | sseq0 4402 | . . . . . . . . . 10 ⊢ ((ran 𝑓 ⊆ 𝐴 ∧ 𝐴 = ∅) → ran 𝑓 = ∅) | |
| 8 | 6, 7 | sylan 580 | . . . . . . . . 9 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝐴 = ∅) → ran 𝑓 = ∅) | 
| 9 | 8 | f1oeq3d 6844 | . . . . . . . 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 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 = ∅)) | |
| 14 | 13 | simprbi 496 | . . . . . . . . 9 ⊢ (◡𝑓:∅–1-1-onto→ω → ω = ∅) | 
| 15 | 14 | eleq2d 2826 | . . . . . . . 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 2946 | . . 3 ⊢ (𝑓:ω–1-1→𝐴 → 𝐴 ≠ ∅) | 
| 21 | 20 | exlimiv 1929 | . 2 ⊢ (∃𝑓 𝑓:ω–1-1→𝐴 → 𝐴 ≠ ∅) | 
| 22 | 1, 21 | syl 17 | 1 ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅) | 
| 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 –1-1→wf1 6557 –1-1-onto→wf1o 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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