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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 4270 | . 2 ⊢ (Ord 𝐵 → Tr 𝐵) | |
2 | trss 4005 | . . . . 5 ⊢ (Tr 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵)) | |
3 | snssi 3634 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
4 | 3 | a1i 9 | . . . . 5 ⊢ (Tr 𝐵 → (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)) |
5 | 2, 4 | jcad 305 | . . . 4 ⊢ (Tr 𝐵 → (𝐴 ∈ 𝐵 → (𝐴 ⊆ 𝐵 ∧ {𝐴} ⊆ 𝐵))) |
6 | unss 3220 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ {𝐴} ⊆ 𝐵) ↔ (𝐴 ∪ {𝐴}) ⊆ 𝐵) | |
7 | 5, 6 | syl6ib 160 | . . 3 ⊢ (Tr 𝐵 → (𝐴 ∈ 𝐵 → (𝐴 ∪ {𝐴}) ⊆ 𝐵)) |
8 | df-suc 4263 | . . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
9 | 8 | sseq1i 3093 | . . 3 ⊢ (suc 𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ {𝐴}) ⊆ 𝐵) |
10 | 7, 9 | syl6ibr 161 | . 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 103 ∈ wcel 1465 ∪ cun 3039 ⊆ wss 3041 {csn 3497 Tr wtr 3996 Ord word 4254 suc csuc 4257 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-uni 3707 df-tr 3997 df-iord 4258 df-suc 4263 |
This theorem is referenced by: ordelsuc 4391 tfrlemibfn 6193 tfr1onlembfn 6209 tfrcllembfn 6222 sucinc2 6310 nndomo 6726 prarloclemn 7275 ennnfonelemhom 11855 ennnfonelemrn 11859 |
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