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 3232 | . 2 ⊢ {𝑥 ∈ On ∣ 𝜑} ⊆ On | |
2 | nfrab1 2649 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ 𝜑} | |
3 | 2 | nfcri 2306 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} |
4 | 3 | nfex 1630 | . . 3 ⊢ Ⅎ𝑥∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑} |
5 | rabid 2645 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑)) | |
6 | elex2 2746 | . . . . 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 2580 | . 2 ⊢ (∃𝑥 ∈ On 𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}) |
10 | onintonm 4501 | . 2 ⊢ (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}) → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) | |
11 | 1, 9, 10 | sylancr 412 | 1 ⊢ (∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1485 ∈ wcel 2141 ∃wrex 2449 {crab 2452 ⊆ wss 3121 ∩ cint 3831 Oncon0 4348 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-int 3832 df-tr 4088 df-iord 4351 df-on 4353 df-suc 4356 |
This theorem is referenced by: cardcl 7158 |
Copyright terms: Public domain | W3C validator |