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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 4515 | . . . . . 6 ⊢ ω ∈ V | |
2 | 1 | snid 3563 | . . . . 5 ⊢ ω ∈ {ω} |
3 | elun2 3249 | . . . . 5 ⊢ (ω ∈ {ω} → ω ∈ (ω ∪ {ω})) | |
4 | breq1 3940 | . . . . . 6 ⊢ (𝑦 = ω → (𝑦 ≼ 𝐴 ↔ ω ≼ 𝐴)) | |
5 | 4 | elrab3 2845 | . . . . 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 2841 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → 𝑧 ∈ (ω ∪ {ω})) | |
9 | elun 3222 | . . . . . . 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 4528 | . . . . . . . 8 ⊢ Ord ω | |
12 | ordelss 4309 | . . . . . . . 8 ⊢ ((Ord ω ∧ 𝑧 ∈ ω) → 𝑧 ⊆ ω) | |
13 | 11, 12 | mpan 421 | . . . . . . 7 ⊢ (𝑧 ∈ ω → 𝑧 ⊆ ω) |
14 | elsni 3550 | . . . . . . . 8 ⊢ (𝑧 ∈ {ω} → 𝑧 = ω) | |
15 | eqimss 3156 | . . . . . . . 8 ⊢ (𝑧 = ω → 𝑧 ⊆ ω) | |
16 | 14, 15 | syl 14 | . . . . . . 7 ⊢ (𝑧 ∈ {ω} → 𝑧 ⊆ ω) |
17 | 13, 16 | jaoi 706 | . . . . . 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 2508 | . . 3 ⊢ (ω ≼ 𝐴 → ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}𝑧 ⊆ ω) |
21 | ssunieq 3777 | . . 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 409 | . 2 ⊢ (ω ≼ 𝐴 → ω = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}) |
23 | 22 | eqcomd 2146 | 1 ⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 698 = wceq 1332 ∈ wcel 1481 ∀wral 2417 {crab 2421 ∪ cun 3074 ⊆ wss 3076 {csn 3532 ∪ cuni 3744 class class class wbr 3937 Ord word 4292 ωcom 4512 ≼ cdom 6641 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-tr 4035 df-iord 4296 df-suc 4301 df-iom 4513 |
This theorem is referenced by: hashinfom 10556 |
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