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Mirrors > Home > ILE Home > Th. List > onintrab2im | GIF version |
Description: An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.) |
Ref | Expression |
---|---|
onintrab2im | ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3146 | . 2 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On | |
2 | nfrab1 2582 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑} | |
3 | 2 | nfcri 2247 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} |
4 | 3 | nfex 1597 | . . 3 ⊢ Ⅎ𝑥∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} |
5 | rabid 2578 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑)) | |
6 | elex2 2671 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}) | |
7 | 5, 6 | sylbir 134 | . . . 4 ⊢ ((𝑥 ∈ On ∧ 𝜑) → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}) |
8 | 7 | ex 114 | . . 3 ⊢ (𝑥 ∈ On → (𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})) |
9 | 4, 8 | rexlimi 2514 | . 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}) |
10 | onintonm 4391 | . 2 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) | |
11 | 1, 9, 10 | sylancr 408 | 1 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1449 ∈ wcel 1461 ∃wrex 2389 {crab 2392 ⊆ wss 3035 ∩ cint 3735 Oncon0 4243 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-uni 3701 df-int 3736 df-tr 3985 df-iord 4246 df-on 4248 df-suc 4251 |
This theorem is referenced by: cardcl 6984 |
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