![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ordsucss | GIF version |
Description: The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.) |
Ref | Expression |
---|---|
ordsucss | ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtr 4378 | . 2 ⊢ (Ord 𝐵 → Tr 𝐵) | |
2 | trss 4110 | . . . . 5 ⊢ (Tr 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵)) | |
3 | snssi 3736 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
4 | 3 | a1i 9 | . . . . 5 ⊢ (Tr 𝐵 → (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)) |
5 | 2, 4 | jcad 307 | . . . 4 ⊢ (Tr 𝐵 → (𝐴 ∈ 𝐵 → (𝐴 ⊆ 𝐵 ∧ {𝐴} ⊆ 𝐵))) |
6 | unss 3309 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ {𝐴} ⊆ 𝐵) ↔ (𝐴 ∪ {𝐴}) ⊆ 𝐵) | |
7 | 5, 6 | imbitrdi 161 | . . 3 ⊢ (Tr 𝐵 → (𝐴 ∈ 𝐵 → (𝐴 ∪ {𝐴}) ⊆ 𝐵)) |
8 | df-suc 4371 | . . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
9 | 8 | sseq1i 3181 | . . 3 ⊢ (suc 𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ {𝐴}) ⊆ 𝐵) |
10 | 7, 9 | syl6ibr 162 | . 2 ⊢ (Tr 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
11 | 1, 10 | syl 14 | 1 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 ∪ cun 3127 ⊆ wss 3129 {csn 3592 Tr wtr 4101 Ord word 4362 suc csuc 4365 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3598 df-uni 3810 df-tr 4102 df-iord 4366 df-suc 4371 |
This theorem is referenced by: ordelsuc 4504 tfrlemibfn 6328 tfr1onlembfn 6344 tfrcllembfn 6357 sucinc2 6446 nndomo 6863 prarloclemn 7497 ennnfonelemhom 12410 ennnfonelemrn 12414 |
Copyright terms: Public domain | W3C validator |