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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 6546 | . . . 4 ⊢ (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1→𝐴) | |
2 | 1 | adantr 271 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) → ∃𝑓 𝑓:ω–1-1→𝐴) |
3 | vex 2636 | . . . . 5 ⊢ 𝑓 ∈ V | |
4 | imaexg 4819 | . . . . 5 ⊢ (𝑓 ∈ V → (𝑓 “ 𝑛) ∈ V) | |
5 | 3, 4 | ax-mp 7 | . . . 4 ⊢ (𝑓 “ 𝑛) ∈ V |
6 | imassrn 4818 | . . . . . 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 5251 | . . . . . . 7 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓:ω⟶𝐴) | |
9 | frn 5204 | . . . . . . 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 | syl5ss 3050 | . . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ 𝑛) ⊆ 𝐴) |
12 | ordom 4449 | . . . . . . . 8 ⊢ Ord ω | |
13 | ordelss 4230 | . . . . . . . 8 ⊢ ((Ord ω ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω) | |
14 | 12, 13 | mpan 416 | . . . . . . 7 ⊢ (𝑛 ∈ ω → 𝑛 ⊆ ω) |
15 | 14 | ad2antlr 474 | . . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → 𝑛 ⊆ ω) |
16 | simplr 498 | . . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → 𝑛 ∈ ω) | |
17 | f1imaeng 6589 | . . . . . 6 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝑛 ⊆ ω ∧ 𝑛 ∈ ω) → (𝑓 “ 𝑛) ≈ 𝑛) | |
18 | 7, 15, 16, 17 | syl3anc 1181 | . . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ 𝑛) ≈ 𝑛) |
19 | 11, 18 | jca 301 | . . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → ((𝑓 “ 𝑛) ⊆ 𝐴 ∧ (𝑓 “ 𝑛) ≈ 𝑛)) |
20 | sseq1 3062 | . . . . . 6 ⊢ (𝑥 = (𝑓 “ 𝑛) → (𝑥 ⊆ 𝐴 ↔ (𝑓 “ 𝑛) ⊆ 𝐴)) | |
21 | breq1 3870 | . . . . . 6 ⊢ (𝑥 = (𝑓 “ 𝑛) → (𝑥 ≈ 𝑛 ↔ (𝑓 “ 𝑛) ≈ 𝑛)) | |
22 | 20, 21 | anbi12d 458 | . . . . 5 ⊢ (𝑥 = (𝑓 “ 𝑛) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ((𝑓 “ 𝑛) ⊆ 𝐴 ∧ (𝑓 “ 𝑛) ≈ 𝑛))) |
23 | 22 | spcegv 2721 | . . . 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 1833 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) |
26 | 25 | ralrimiva 2458 | 1 ⊢ (ω ≼ 𝐴 → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∃wex 1433 ∈ wcel 1445 ∀wral 2370 Vcvv 2633 ⊆ wss 3013 class class class wbr 3867 Ord word 4213 ωcom 4433 ran crn 4468 “ cima 4470 ⟶wf 5045 –1-1→wf1 5046 ≈ cen 6535 ≼ cdom 6536 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-iord 4217 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-er 6332 df-en 6538 df-dom 6539 |
This theorem is referenced by: (None) |
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