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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 4371 | . . . . . 6 ⊢ ω ∈ V | |
2 | 1 | snid 3449 | . . . . 5 ⊢ ω ∈ {ω} |
3 | elun2 3152 | . . . . 5 ⊢ (ω ∈ {ω} → ω ∈ (ω ∪ {ω})) | |
4 | breq1 3814 | . . . . . 6 ⊢ (𝑦 = ω → (𝑦 ≼ 𝐴 ↔ ω ≼ 𝐴)) | |
5 | 4 | elrab3 2760 | . . . . 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 131 | . . 3 ⊢ (ω ≼ 𝐴 → ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}) |
8 | elrabi 2756 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → 𝑧 ∈ (ω ∪ {ω})) | |
9 | elun 3125 | . . . . . . 7 ⊢ (𝑧 ∈ (ω ∪ {ω}) ↔ (𝑧 ∈ ω ∨ 𝑧 ∈ {ω})) | |
10 | 8, 9 | sylib 120 | . . . . . 6 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → (𝑧 ∈ ω ∨ 𝑧 ∈ {ω})) |
11 | ordom 4384 | . . . . . . . 8 ⊢ Ord ω | |
12 | ordelss 4170 | . . . . . . . 8 ⊢ ((Ord ω ∧ 𝑧 ∈ ω) → 𝑧 ⊆ ω) | |
13 | 11, 12 | mpan 415 | . . . . . . 7 ⊢ (𝑧 ∈ ω → 𝑧 ⊆ ω) |
14 | elsni 3440 | . . . . . . . 8 ⊢ (𝑧 ∈ {ω} → 𝑧 = ω) | |
15 | eqimss 3062 | . . . . . . . 8 ⊢ (𝑧 = ω → 𝑧 ⊆ ω) | |
16 | 14, 15 | syl 14 | . . . . . . 7 ⊢ (𝑧 ∈ {ω} → 𝑧 ⊆ ω) |
17 | 13, 16 | jaoi 669 | . . . . . 6 ⊢ ((𝑧 ∈ ω ∨ 𝑧 ∈ {ω}) → 𝑧 ⊆ ω) |
18 | 10, 17 | syl 14 | . . . . 5 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → 𝑧 ⊆ ω) |
19 | 18 | adantl 271 | . . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}) → 𝑧 ⊆ ω) |
20 | 19 | ralrimiva 2440 | . . 3 ⊢ (ω ≼ 𝐴 → ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}𝑧 ⊆ ω) |
21 | ssunieq 3660 | . . 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 403 | . 2 ⊢ (ω ≼ 𝐴 → ω = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}) |
23 | 22 | eqcomd 2088 | 1 ⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∨ wo 662 = wceq 1285 ∈ wcel 1434 ∀wral 2353 {crab 2357 ∪ cun 2982 ⊆ wss 2984 {csn 3422 ∪ cuni 3627 class class class wbr 3811 Ord word 4153 ωcom 4368 ≼ cdom 6386 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-iinf 4366 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-tr 3902 df-iord 4157 df-suc 4162 df-iom 4369 |
This theorem is referenced by: hashinfom 10021 |
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