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Mirrors > Home > ILE Home > Th. List > hashinfuni | GIF version |
Description: The ordinal size of an infinite set is ω. (Contributed by Jim Kingdon, 20-Feb-2022.) |
Ref | Expression |
---|---|
hashinfuni | ⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omex 4477 | . . . . . 6 ⊢ ω ∈ V | |
2 | 1 | snid 3526 | . . . . 5 ⊢ ω ∈ {ω} |
3 | elun2 3214 | . . . . 5 ⊢ (ω ∈ {ω} → ω ∈ (ω ∪ {ω})) | |
4 | breq1 3902 | . . . . . 6 ⊢ (𝑦 = ω → (𝑦 ≼ 𝐴 ↔ ω ≼ 𝐴)) | |
5 | 4 | elrab3 2814 | . . . . 5 ⊢ (ω ∈ (ω ∪ {ω}) → (ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} ↔ ω ≼ 𝐴)) |
6 | 2, 3, 5 | mp2b 8 | . . . 4 ⊢ (ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} ↔ ω ≼ 𝐴) |
7 | 6 | biimpri 132 | . . 3 ⊢ (ω ≼ 𝐴 → ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}) |
8 | elrabi 2810 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → 𝑧 ∈ (ω ∪ {ω})) | |
9 | elun 3187 | . . . . . . 7 ⊢ (𝑧 ∈ (ω ∪ {ω}) ↔ (𝑧 ∈ ω ∨ 𝑧 ∈ {ω})) | |
10 | 8, 9 | sylib 121 | . . . . . 6 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → (𝑧 ∈ ω ∨ 𝑧 ∈ {ω})) |
11 | ordom 4490 | . . . . . . . 8 ⊢ Ord ω | |
12 | ordelss 4271 | . . . . . . . 8 ⊢ ((Ord ω ∧ 𝑧 ∈ ω) → 𝑧 ⊆ ω) | |
13 | 11, 12 | mpan 420 | . . . . . . 7 ⊢ (𝑧 ∈ ω → 𝑧 ⊆ ω) |
14 | elsni 3515 | . . . . . . . 8 ⊢ (𝑧 ∈ {ω} → 𝑧 = ω) | |
15 | eqimss 3121 | . . . . . . . 8 ⊢ (𝑧 = ω → 𝑧 ⊆ ω) | |
16 | 14, 15 | syl 14 | . . . . . . 7 ⊢ (𝑧 ∈ {ω} → 𝑧 ⊆ ω) |
17 | 13, 16 | jaoi 690 | . . . . . 6 ⊢ ((𝑧 ∈ ω ∨ 𝑧 ∈ {ω}) → 𝑧 ⊆ ω) |
18 | 10, 17 | syl 14 | . . . . 5 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → 𝑧 ⊆ ω) |
19 | 18 | adantl 275 | . . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}) → 𝑧 ⊆ ω) |
20 | 19 | ralrimiva 2482 | . . 3 ⊢ (ω ≼ 𝐴 → ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}𝑧 ⊆ ω) |
21 | ssunieq 3739 | . . 3 ⊢ ((ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} ∧ ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}𝑧 ⊆ ω) → ω = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}) | |
22 | 7, 20, 21 | syl2anc 408 | . 2 ⊢ (ω ≼ 𝐴 → ω = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}) |
23 | 22 | eqcomd 2123 | 1 ⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 682 = wceq 1316 ∈ wcel 1465 ∀wral 2393 {crab 2397 ∪ cun 3039 ⊆ wss 3041 {csn 3497 ∪ cuni 3706 class class class wbr 3899 Ord word 4254 ωcom 4474 ≼ 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-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-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 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-br 3900 df-tr 3997 df-iord 4258 df-suc 4263 df-iom 4475 |
This theorem is referenced by: hashinfom 10492 |
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