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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. (Contributed by NM, 21-Jun-1998.) |
Ref | Expression |
---|---|
ordsucss | ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordelord 6216 | . . . . 5 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → Ord 𝐴) | |
2 | ordnbtwn 6284 | . . . . . . . 8 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)) | |
3 | imnan 402 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝐴) ↔ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)) | |
4 | 2, 3 | sylibr 236 | . . . . . . 7 ⊢ (Ord 𝐴 → (𝐴 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝐴)) |
5 | 4 | adantr 483 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 → ¬ 𝐵 ∈ suc 𝐴)) |
6 | ordsuc 7532 | . . . . . . 7 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
7 | ordtri1 6227 | . . . . . . 7 ⊢ ((Ord suc 𝐴 ∧ Ord 𝐵) → (suc 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ suc 𝐴)) | |
8 | 6, 7 | sylanb 583 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (suc 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ suc 𝐴)) |
9 | 5, 8 | sylibrd 261 | . . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
10 | 1, 9 | sylan 582 | . . . 4 ⊢ (((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵)) |
11 | 10 | exp31 422 | . . 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 208 ∧ wa 398 ∈ wcel 2113 ⊆ wss 3939 Ord word 6193 suc csuc 6196 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-tr 5176 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-ord 6197 df-on 6198 df-suc 6200 |
This theorem is referenced by: ordelsuc 7538 ordsucelsuc 7540 orduniorsuc 7548 tfindsg2 7579 oaordi 8175 oawordeulem 8183 omeulem2 8212 oeworde 8222 oelimcl 8229 oeeui 8231 nnaordi 8247 nnawordex 8266 oaabs2 8275 omxpenlem 8621 inf3lem5 9098 cantnflt 9138 cantnflem1d 9154 cnfcom 9166 r1ordg 9210 rankr1ag 9234 cfslb2n 9693 cfsmolem 9695 fin23lem26 9750 isf32lem3 9780 ttukeylem7 9940 indpi 10332 nolesgn2ores 33183 nosupres 33211 nosupbnd1lem1 33212 nosupbnd2 33220 |
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