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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 4476 | . . 3 ⊢ (𝑋 ⊊ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋)) | |
2 | 1 | adantr 480 | . 2 ⊢ ((𝑋 ⊊ 𝐴 ∧ 𝑋 ≈ 𝐴) → ∃𝑦(𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋)) |
3 | eldif 3972 | . . . 4 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑋) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑋)) | |
4 | pssss 4107 | . . . . . 6 ⊢ (𝑋 ⊊ 𝐴 → 𝑋 ⊆ 𝐴) | |
5 | bren 8993 | . . . . . . . 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 3481 | . . . . . . . . . . . . 13 ⊢ 𝑓 ∈ V | |
11 | 10 | rnex 7932 | . . . . . . . . . . . 12 ⊢ ran 𝑓 ∈ V |
12 | 9, 11 | eqeltrrdi 2847 | . . . . . . . . . . 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 2734 | . . . . . . . . . . . 12 ⊢ (rec(◡𝑓, 𝑦) ↾ ω) = (rec(◡𝑓, 𝑦) ↾ ω) | |
16 | 13, 6, 14, 15 | infpssrlem5 10344 | . . . . . . . . . . 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 1930 | . . . . . . . 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 1927 | . 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 1536 ∃wex 1775 ∈ wcel 2105 Vcvv 3477 ∖ cdif 3959 ⊆ wss 3962 ⊊ wpss 3963 class class class wbr 5147 ◡ccnv 5687 ran crn 5689 ↾ cres 5690 –onto→wfo 6560 –1-1-onto→wf1o 6561 ωcom 7886 reccrdg 8447 ≈ cen 8980 ≼ cdom 8981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-en 8984 df-dom 8985 |
This theorem is referenced by: isfin4-2 10351 |
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