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Mirrors > Home > MPE Home > Th. List > infn0 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
infn0 | ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomi 8989 | . 2 ⊢ (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1→𝐴) | |
2 | peano1 7900 | . . . . . 6 ⊢ ∅ ∈ ω | |
3 | f1f1orn 6854 | . . . . . . . . 9 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓:ω–1-1-onto→ran 𝑓) | |
4 | 3 | adantr 479 | . . . . . . . 8 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝐴 = ∅) → 𝑓:ω–1-1-onto→ran 𝑓) |
5 | f1f 6798 | . . . . . . . . . . 11 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓:ω⟶𝐴) | |
6 | 5 | frnd 6736 | . . . . . . . . . 10 ⊢ (𝑓:ω–1-1→𝐴 → ran 𝑓 ⊆ 𝐴) |
7 | sseq0 4404 | . . . . . . . . . 10 ⊢ ((ran 𝑓 ⊆ 𝐴 ∧ 𝐴 = ∅) → ran 𝑓 = ∅) | |
8 | 6, 7 | sylan 578 | . . . . . . . . 9 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝐴 = ∅) → ran 𝑓 = ∅) |
9 | 8 | f1oeq3d 6840 | . . . . . . . 8 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝐴 = ∅) → (𝑓:ω–1-1-onto→ran 𝑓 ↔ 𝑓:ω–1-1-onto→∅)) |
10 | 4, 9 | mpbid 231 | . . . . . . 7 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝐴 = ∅) → 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 = ∅)) | |
14 | 13 | simprbi 495 | . . . . . . . . 9 ⊢ (◡𝑓:∅–1-1-onto→ω → ω = ∅) |
15 | 14 | eleq2d 2812 | . . . . . . . 8 ⊢ (◡𝑓:∅–1-1-onto→ω → (∅ ∈ ω ↔ ∅ ∈ ∅)) |
16 | 12, 15 | mtbiri 326 | . . . . . . 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 199 | . . . . 5 ⊢ ¬ (𝑓:ω–1-1→𝐴 ∧ 𝐴 = ∅) |
19 | 18 | imnani 399 | . . . 4 ⊢ (𝑓:ω–1-1→𝐴 → ¬ 𝐴 = ∅) |
20 | 19 | neqned 2937 | . . 3 ⊢ (𝑓:ω–1-1→𝐴 → 𝐴 ≠ ∅) |
21 | 20 | exlimiv 1926 | . 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 394 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2930 ⊆ wss 3947 ∅c0 4325 class class class wbr 5153 ◡ccnv 5681 ran crn 5683 –1-1→wf1 6551 –1-1-onto→wf1o 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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