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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 4502 | . . . . . 6 ⊢ ω ∈ V | |
2 | 1 | snid 3551 | . . . . 5 ⊢ ω ∈ {ω} |
3 | elun2 3239 | . . . . 5 ⊢ (ω ∈ {ω} → ω ∈ (ω ∪ {ω})) | |
4 | breq1 3927 | . . . . . 6 ⊢ (𝑦 = ω → (𝑦 ≼ 𝐴 ↔ ω ≼ 𝐴)) | |
5 | 4 | elrab3 2836 | . . . . 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 2832 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → 𝑧 ∈ (ω ∪ {ω})) | |
9 | elun 3212 | . . . . . . 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 4515 | . . . . . . . 8 ⊢ Ord ω | |
12 | ordelss 4296 | . . . . . . . 8 ⊢ ((Ord ω ∧ 𝑧 ∈ ω) → 𝑧 ⊆ ω) | |
13 | 11, 12 | mpan 420 | . . . . . . 7 ⊢ (𝑧 ∈ ω → 𝑧 ⊆ ω) |
14 | elsni 3540 | . . . . . . . 8 ⊢ (𝑧 ∈ {ω} → 𝑧 = ω) | |
15 | eqimss 3146 | . . . . . . . 8 ⊢ (𝑧 = ω → 𝑧 ⊆ ω) | |
16 | 14, 15 | syl 14 | . . . . . . 7 ⊢ (𝑧 ∈ {ω} → 𝑧 ⊆ ω) |
17 | 13, 16 | jaoi 705 | . . . . . 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 2503 | . . 3 ⊢ (ω ≼ 𝐴 → ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}𝑧 ⊆ ω) |
21 | ssunieq 3764 | . . 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 2143 | 1 ⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 697 = wceq 1331 ∈ wcel 1480 ∀wral 2414 {crab 2418 ∪ cun 3064 ⊆ wss 3066 {csn 3522 ∪ cuni 3731 class class class wbr 3924 Ord word 4279 ωcom 4499 ≼ cdom 6626 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-tr 4022 df-iord 4283 df-suc 4288 df-iom 4500 |
This theorem is referenced by: hashinfom 10517 |
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