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Mirrors > Home > ILE Home > Th. List > isinfinf | GIF version |
Description: An infinite set contains subsets of arbitrarily large finite cardinality. (Contributed by Jim Kingdon, 15-Jun-2022.) |
Ref | Expression |
---|---|
isinfinf | ⊢ (ω ≼ 𝐴 → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomi 6611 | . . . 4 ⊢ (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1→𝐴) | |
2 | 1 | adantr 274 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) → ∃𝑓 𝑓:ω–1-1→𝐴) |
3 | vex 2663 | . . . . 5 ⊢ 𝑓 ∈ V | |
4 | imaexg 4863 | . . . . 5 ⊢ (𝑓 ∈ V → (𝑓 “ 𝑛) ∈ V) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (𝑓 “ 𝑛) ∈ V |
6 | imassrn 4862 | . . . . . 6 ⊢ (𝑓 “ 𝑛) ⊆ ran 𝑓 | |
7 | simpr 109 | . . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → 𝑓:ω–1-1→𝐴) | |
8 | f1f 5298 | . . . . . . 7 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓:ω⟶𝐴) | |
9 | frn 5251 | . . . . . . 7 ⊢ (𝑓:ω⟶𝐴 → ran 𝑓 ⊆ 𝐴) | |
10 | 7, 8, 9 | 3syl 17 | . . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → ran 𝑓 ⊆ 𝐴) |
11 | 6, 10 | sstrid 3078 | . . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ 𝑛) ⊆ 𝐴) |
12 | ordom 4490 | . . . . . . . 8 ⊢ Ord ω | |
13 | ordelss 4271 | . . . . . . . 8 ⊢ ((Ord ω ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω) | |
14 | 12, 13 | mpan 420 | . . . . . . 7 ⊢ (𝑛 ∈ ω → 𝑛 ⊆ ω) |
15 | 14 | ad2antlr 480 | . . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → 𝑛 ⊆ ω) |
16 | simplr 504 | . . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → 𝑛 ∈ ω) | |
17 | f1imaeng 6654 | . . . . . 6 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝑛 ⊆ ω ∧ 𝑛 ∈ ω) → (𝑓 “ 𝑛) ≈ 𝑛) | |
18 | 7, 15, 16, 17 | syl3anc 1201 | . . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ 𝑛) ≈ 𝑛) |
19 | 11, 18 | jca 304 | . . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → ((𝑓 “ 𝑛) ⊆ 𝐴 ∧ (𝑓 “ 𝑛) ≈ 𝑛)) |
20 | sseq1 3090 | . . . . . 6 ⊢ (𝑥 = (𝑓 “ 𝑛) → (𝑥 ⊆ 𝐴 ↔ (𝑓 “ 𝑛) ⊆ 𝐴)) | |
21 | breq1 3902 | . . . . . 6 ⊢ (𝑥 = (𝑓 “ 𝑛) → (𝑥 ≈ 𝑛 ↔ (𝑓 “ 𝑛) ≈ 𝑛)) | |
22 | 20, 21 | anbi12d 464 | . . . . 5 ⊢ (𝑥 = (𝑓 “ 𝑛) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ((𝑓 “ 𝑛) ⊆ 𝐴 ∧ (𝑓 “ 𝑛) ≈ 𝑛))) |
23 | 22 | spcegv 2748 | . . . 4 ⊢ ((𝑓 “ 𝑛) ∈ V → (((𝑓 “ 𝑛) ⊆ 𝐴 ∧ (𝑓 “ 𝑛) ≈ 𝑛) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))) |
24 | 5, 19, 23 | mpsyl 65 | . . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) |
25 | 2, 24 | exlimddv 1854 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) |
26 | 25 | ralrimiva 2482 | 1 ⊢ (ω ≼ 𝐴 → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∃wex 1453 ∈ wcel 1465 ∀wral 2393 Vcvv 2660 ⊆ wss 3041 class class class wbr 3899 Ord word 4254 ωcom 4474 ran crn 4510 “ cima 4512 ⟶wf 5089 –1-1→wf1 5090 ≈ cen 6600 ≼ cdom 6601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-er 6397 df-en 6603 df-dom 6604 |
This theorem is referenced by: (None) |
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