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| Mirrors > Home > MPE Home > Th. List > infpssr | Structured version Visualization version GIF version | ||
| Description: Dedekind infinity implies existence of a denumerable subset: take a single point witnessing the proper subset relation and iterate the embedding. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) |
| Ref | Expression |
|---|---|
| infpssr | ⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pssnel 4471 | . . 3 ⊢ (𝑋 ⊊ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋)) | |
| 2 | 1 | adantr 480 | . 2 ⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋)) |
| 3 | eldif 3961 | . . . 4 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑋) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋)) | |
| 4 | pssss 4098 | . . . . . 6 ⊢ (𝑋 ⊊ 𝐴 → 𝑋 ⊆ 𝐴) | |
| 5 | bren 8995 | . . . . . . . 8 ⊢ (𝑋 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝑋–1-1-onto→𝐴) | |
| 6 | simpr 484 | . . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → 𝑓:𝑋–1-1-onto→𝐴) | |
| 7 | f1ofo 6855 | . . . . . . . . . . . . 13 ⊢ (𝑓:𝑋–1-1-onto→𝐴 → 𝑓:𝑋–onto→𝐴) | |
| 8 | forn 6823 | . . . . . . . . . . . . 13 ⊢ (𝑓:𝑋–onto→𝐴 → ran 𝑓 = 𝐴) | |
| 9 | 6, 7, 8 | 3syl 18 | . . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → ran 𝑓 = 𝐴) |
| 10 | vex 3484 | . . . . . . . . . . . . 13 ⊢ 𝑓 ∈ V | |
| 11 | 10 | rnex 7932 | . . . . . . . . . . . 12 ⊢ ran 𝑓 ∈ V |
| 12 | 9, 11 | eqeltrrdi 2850 | . . . . . . . . . . 11 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → 𝐴 ∈ V) |
| 13 | simplr 769 | . . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → 𝑋 ⊆ 𝐴) | |
| 14 | simpll 767 | . . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → 𝑦 ∈ (𝐴 ∖ 𝑋)) | |
| 15 | eqid 2737 | . . . . . . . . . . . 12 ⊢ (rec(◡𝑓, 𝑦) ↾ ω) = (rec(◡𝑓, 𝑦) ↾ ω) | |
| 16 | 13, 6, 14, 15 | infpssrlem5 10347 | . . . . . . . . . . 11 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → (𝐴 ∈ V → ω ≼ 𝐴)) |
| 17 | 12, 16 | mpd 15 | . . . . . . . . . 10 ⊢ (((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) ∧ 𝑓:𝑋–1-1-onto→𝐴) → ω ≼ 𝐴) |
| 18 | 17 | ex 412 | . . . . . . . . 9 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) → (𝑓:𝑋–1-1-onto→𝐴 → ω ≼ 𝐴)) |
| 19 | 18 | exlimdv 1933 | . . . . . . . 8 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) → (∃𝑓 𝑓:𝑋–1-1-onto→𝐴 → ω ≼ 𝐴)) |
| 20 | 5, 19 | biimtrid 242 | . . . . . . 7 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝑋) ∧ 𝑋 ⊆ 𝐴) → (𝑋 ≈ 𝐴 → ω ≼ 𝐴)) |
| 21 | 20 | ex 412 | . . . . . 6 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑋) → (𝑋 ⊆ 𝐴 → (𝑋 ≈ 𝐴 → ω ≼ 𝐴))) |
| 22 | 4, 21 | syl5 34 | . . . . 5 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑋) → (𝑋 ⊊ 𝐴 → (𝑋 ≈ 𝐴 → ω ≼ 𝐴))) |
| 23 | 22 | impd 410 | . . . 4 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑋) → ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴)) |
| 24 | 3, 23 | sylbir 235 | . . 3 ⊢ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋) → ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴)) |
| 25 | 24 | exlimiv 1930 | . 2 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋) → ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴)) |
| 26 | 2, 25 | mpcom 38 | 1 ⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ω ≼ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 ⊊ wpss 3952 class class class wbr 5143 ◡ccnv 5684 ran crn 5686 ↾ cres 5687 –onto→wfo 6559 –1-1-onto→wf1o 6560 ωcom 7887 reccrdg 8449 ≈ cen 8982 ≼ cdom 8983 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-en 8986 df-dom 8987 |
| This theorem is referenced by: isfin4-2 10354 |
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