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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 6651 | . . . 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 2692 | . . . . 5 ⊢ 𝑓 ∈ V | |
4 | imaexg 4901 | . . . . 5 ⊢ (𝑓 ∈ V → (𝑓 “ 𝑛) ∈ V) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (𝑓 “ 𝑛) ∈ V |
6 | imassrn 4900 | . . . . . 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 5336 | . . . . . . 7 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓:ω⟶𝐴) | |
9 | frn 5289 | . . . . . . 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 3113 | . . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ 𝑛) ⊆ 𝐴) |
12 | ordom 4528 | . . . . . . . 8 ⊢ Ord ω | |
13 | ordelss 4309 | . . . . . . . 8 ⊢ ((Ord ω ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω) | |
14 | 12, 13 | mpan 421 | . . . . . . 7 ⊢ (𝑛 ∈ ω → 𝑛 ⊆ ω) |
15 | 14 | ad2antlr 481 | . . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → 𝑛 ⊆ ω) |
16 | simplr 520 | . . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → 𝑛 ∈ ω) | |
17 | f1imaeng 6694 | . . . . . 6 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝑛 ⊆ ω ∧ 𝑛 ∈ ω) → (𝑓 “ 𝑛) ≈ 𝑛) | |
18 | 7, 15, 16, 17 | syl3anc 1217 | . . . . 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 3125 | . . . . . 6 ⊢ (𝑥 = (𝑓 “ 𝑛) → (𝑥 ⊆ 𝐴 ↔ (𝑓 “ 𝑛) ⊆ 𝐴)) | |
21 | breq1 3940 | . . . . . 6 ⊢ (𝑥 = (𝑓 “ 𝑛) → (𝑥 ≈ 𝑛 ↔ (𝑓 “ 𝑛) ≈ 𝑛)) | |
22 | 20, 21 | anbi12d 465 | . . . . 5 ⊢ (𝑥 = (𝑓 “ 𝑛) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ((𝑓 “ 𝑛) ⊆ 𝐴 ∧ (𝑓 “ 𝑛) ≈ 𝑛))) |
23 | 22 | spcegv 2777 | . . . 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 1871 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) |
26 | 25 | ralrimiva 2508 | 1 ⊢ (ω ≼ 𝐴 → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∃wex 1469 ∈ wcel 1481 ∀wral 2417 Vcvv 2689 ⊆ wss 3076 class class class wbr 3937 Ord word 4292 ωcom 4512 ran crn 4548 “ cima 4550 ⟶wf 5127 –1-1→wf1 5128 ≈ cen 6640 ≼ cdom 6641 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-er 6437 df-en 6643 df-dom 6644 |
This theorem is referenced by: (None) |
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