Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3182 | . 2 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On | |
2 | nfrab1 2610 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑} | |
3 | 2 | nfcri 2275 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} |
4 | 3 | nfex 1616 | . . 3 ⊢ Ⅎ𝑥∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} |
5 | rabid 2606 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑)) | |
6 | elex2 2702 | . . . . 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 2542 | . 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}) |
10 | onintonm 4433 | . 2 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) | |
11 | 1, 9, 10 | sylancr 410 | 1 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1468 ∈ wcel 1480 ∃wrex 2417 {crab 2420 ⊆ wss 3071 ∩ cint 3771 Oncon0 4285 |
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-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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-tr 4027 df-iord 4288 df-on 4290 df-suc 4293 |
This theorem is referenced by: cardcl 7037 |
Copyright terms: Public domain | W3C validator |