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| Mirrors > Home > MPE Home > Th. List > ordsucss | Structured version Visualization version GIF version | ||
| Description: The successor of an element of an ordinal class is a subset of it. Lemma 1.14 of [Schloeder] p. 2. (Contributed by NM, 21-Jun-1998.) |
| Ref | Expression |
|---|---|
| ordsucss | ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ordelord 6406 | . . . . 5 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → Ord 𝐴) | |
| 2 | ordnbtwn 6477 | . . . . . . . 8 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)) | |
| 3 | imnan 399 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝐴) ↔ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)) | |
| 4 | 2, 3 | sylibr 234 | . . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝐴)) |
| 5 | 4 | adantr 480 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝐴)) |
| 6 | ordsuc 7833 | . . . . . . 7 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
| 7 | ordtri1 6417 | . . . . . . 7 ⊢ ((Ord suc 𝐴 ∧ Ord 𝐵) → (suc 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ suc 𝐴)) | |
| 8 | 6, 7 | sylanb 581 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (suc 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ suc 𝐴)) |
| 9 | 5, 8 | sylibrd 259 | . . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
| 10 | 1, 9 | sylan 580 | . . . 4 ⊢ (((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
| 11 | 10 | exp31 419 | . . 3 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)))) |
| 12 | 11 | pm2.43b 55 | . 2 ⊢ (𝐴 ∈ 𝐵 → (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))) |
| 13 | 12 | pm2.43b 55 | 1 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 Ord word 6383 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-suc 6390 |
| This theorem is referenced by: ordelsuc 7840 ordsucelsuc 7842 orduniorsuc 7850 tfindsg2 7883 oaordi 8584 oawordeulem 8592 omeulem2 8621 oeworde 8631 oelimcl 8638 oeeui 8640 nnaordi 8656 nnawordex 8675 oaabs2 8687 omxpenlem 9113 inf3lem5 9672 cantnflt 9712 cantnflem1d 9728 cnfcom 9740 r1ordg 9818 rankr1ag 9842 cfslb2n 10308 cfsmolem 10310 fin23lem26 10365 isf32lem3 10395 ttukeylem7 10555 indpi 10947 nolesgn2ores 27717 nogesgn1ores 27719 nosupbday 27750 nosupres 27752 nosupbnd1lem1 27753 nosupbnd2 27761 noinfbday 27765 noinfres 27767 noinfbnd1lem1 27768 noinfbnd2 27776 onsucss 43279 omabs2 43345 onsucunifi 43383 nadd1suc 43405 |
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