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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 4606 | . . . . . 6 ⊢ ω ∈ V | |
2 | 1 | snid 3637 | . . . . 5 ⊢ ω ∈ {ω} |
3 | elun2 3317 | . . . . 5 ⊢ (ω ∈ {ω} → ω ∈ (ω ∪ {ω})) | |
4 | breq1 4020 | . . . . . 6 ⊢ (𝑦 = ω → (𝑦 ≼ 𝐴 ↔ ω ≼ 𝐴)) | |
5 | 4 | elrab3 2908 | . . . . 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 133 | . . 3 ⊢ (ω ≼ 𝐴 → ω ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}) |
8 | elrabi 2904 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → 𝑧 ∈ (ω ∪ {ω})) | |
9 | elun 3290 | . . . . . . 7 ⊢ (𝑧 ∈ (ω ∪ {ω}) ↔ (𝑧 ∈ ω ∨ 𝑧 ∈ {ω})) | |
10 | 8, 9 | sylib 122 | . . . . . 6 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → (𝑧 ∈ ω ∨ 𝑧 ∈ {ω})) |
11 | ordom 4620 | . . . . . . . 8 ⊢ Ord ω | |
12 | ordelss 4393 | . . . . . . . 8 ⊢ ((Ord ω ∧ 𝑧 ∈ ω) → 𝑧 ⊆ ω) | |
13 | 11, 12 | mpan 424 | . . . . . . 7 ⊢ (𝑧 ∈ ω → 𝑧 ⊆ ω) |
14 | elsni 3624 | . . . . . . . 8 ⊢ (𝑧 ∈ {ω} → 𝑧 = ω) | |
15 | eqimss 3223 | . . . . . . . 8 ⊢ (𝑧 = ω → 𝑧 ⊆ ω) | |
16 | 14, 15 | syl 14 | . . . . . . 7 ⊢ (𝑧 ∈ {ω} → 𝑧 ⊆ ω) |
17 | 13, 16 | jaoi 717 | . . . . . 6 ⊢ ((𝑧 ∈ ω ∨ 𝑧 ∈ {ω}) → 𝑧 ⊆ ω) |
18 | 10, 17 | syl 14 | . . . . 5 ⊢ (𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} → 𝑧 ⊆ ω) |
19 | 18 | adantl 277 | . . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}) → 𝑧 ⊆ ω) |
20 | 19 | ralrimiva 2562 | . . 3 ⊢ (ω ≼ 𝐴 → ∀𝑧 ∈ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}𝑧 ⊆ ω) |
21 | ssunieq 3856 | . . 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 411 | . 2 ⊢ (ω ≼ 𝐴 → ω = ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴}) |
23 | 22 | eqcomd 2194 | 1 ⊢ (ω ≼ 𝐴 → ∪ {𝑦 ∈ (ω ∪ {ω}) ∣ 𝑦 ≼ 𝐴} = ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∨ wo 709 = wceq 1363 ∈ wcel 2159 ∀wral 2467 {crab 2471 ∪ cun 3141 ⊆ wss 3143 {csn 3606 ∪ cuni 3823 class class class wbr 4017 Ord word 4376 ωcom 4603 ≼ cdom 6756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-nul 4143 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-iinf 4601 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ral 2472 df-rex 2473 df-rab 2476 df-v 2753 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-nul 3437 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-br 4018 df-tr 4116 df-iord 4380 df-suc 4385 df-iom 4604 |
This theorem is referenced by: hashinfom 10775 |
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