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Mirrors > Home > ILE Home > Th. List > onsucmin | GIF version |
Description: The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.) |
Ref | Expression |
---|---|
onsucmin | ⊢ (A ∈ On → suc A = ∩ {x ∈ On ∣ A ∈ x}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 4078 | . . . . 5 ⊢ (x ∈ On → Ord x) | |
2 | ordelsuc 4197 | . . . . 5 ⊢ ((A ∈ On ∧ Ord x) → (A ∈ x ↔ suc A ⊆ x)) | |
3 | 1, 2 | sylan2 270 | . . . 4 ⊢ ((A ∈ On ∧ x ∈ On) → (A ∈ x ↔ suc A ⊆ x)) |
4 | 3 | rabbidva 2542 | . . 3 ⊢ (A ∈ On → {x ∈ On ∣ A ∈ x} = {x ∈ On ∣ suc A ⊆ x}) |
5 | 4 | inteqd 3611 | . 2 ⊢ (A ∈ On → ∩ {x ∈ On ∣ A ∈ x} = ∩ {x ∈ On ∣ suc A ⊆ x}) |
6 | sucelon 4195 | . . 3 ⊢ (A ∈ On ↔ suc A ∈ On) | |
7 | intmin 3626 | . . 3 ⊢ (suc A ∈ On → ∩ {x ∈ On ∣ suc A ⊆ x} = suc A) | |
8 | 6, 7 | sylbi 114 | . 2 ⊢ (A ∈ On → ∩ {x ∈ On ∣ suc A ⊆ x} = suc A) |
9 | 5, 8 | eqtr2d 2070 | 1 ⊢ (A ∈ On → suc A = ∩ {x ∈ On ∣ A ∈ x}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ∈ wcel 1390 {crab 2304 ⊆ wss 2911 ∩ cint 3606 Ord word 4065 Oncon0 4066 suc csuc 4068 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-uni 3572 df-int 3607 df-tr 3846 df-iord 4069 df-on 4071 df-suc 4074 |
This theorem is referenced by: (None) |
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