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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 6707 | . . . 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 2725 | . . . . 5 ⊢ 𝑓 ∈ V | |
4 | imaexg 4953 | . . . . 5 ⊢ (𝑓 ∈ V → (𝑓 “ 𝑛) ∈ V) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (𝑓 “ 𝑛) ∈ V |
6 | imassrn 4952 | . . . . . 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 5388 | . . . . . . 7 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓:ω⟶𝐴) | |
9 | frn 5341 | . . . . . . 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 3149 | . . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ 𝑛) ⊆ 𝐴) |
12 | ordom 4579 | . . . . . . . 8 ⊢ Ord ω | |
13 | ordelss 4352 | . . . . . . . 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 6750 | . . . . . 6 ⊢ ((𝑓:ω–1-1→𝐴 ∧ 𝑛 ⊆ ω ∧ 𝑛 ∈ ω) → (𝑓 “ 𝑛) ≈ 𝑛) | |
18 | 7, 15, 16, 17 | syl3anc 1227 | . . . . 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 3161 | . . . . . 6 ⊢ (𝑥 = (𝑓 “ 𝑛) → (𝑥 ⊆ 𝐴 ↔ (𝑓 “ 𝑛) ⊆ 𝐴)) | |
21 | breq1 3980 | . . . . . 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 2810 | . . . 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 1885 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝑛 ∈ ω) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) |
26 | 25 | ralrimiva 2537 | 1 ⊢ (ω ≼ 𝐴 → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∃wex 1479 ∈ wcel 2135 ∀wral 2442 Vcvv 2722 ⊆ wss 3112 class class class wbr 3977 Ord word 4335 ωcom 4562 ran crn 4600 “ cima 4602 ⟶wf 5179 –1-1→wf1 5180 ≈ cen 6696 ≼ cdom 6697 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-iinf 4560 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-id 4266 df-iord 4339 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-er 6493 df-en 6699 df-dom 6700 |
This theorem is referenced by: (None) |
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