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Mirrors > Home > MPE Home > Th. List > ordsucOLD | Structured version Visualization version GIF version |
Description: Obsolete version of ordsuc 7753 as of 6-Jan-2025. (Contributed by NM, 3-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ordsucOLD | ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elong 6330 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴)) | |
2 | onsuc 7751 | . . . . 5 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
3 | eloni 6332 | . . . . 5 ⊢ (suc 𝐴 ∈ On → Ord suc 𝐴) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐴 ∈ On → Ord suc 𝐴) |
5 | 1, 4 | syl6bir 254 | . . 3 ⊢ (𝐴 ∈ V → (Ord 𝐴 → Ord suc 𝐴)) |
6 | sucidg 6403 | . . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴) | |
7 | ordelord 6344 | . . . . 5 ⊢ ((Ord suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → Ord 𝐴) | |
8 | 7 | ex 414 | . . . 4 ⊢ (Ord suc 𝐴 → (𝐴 ∈ suc 𝐴 → Ord 𝐴)) |
9 | 6, 8 | syl5com 31 | . . 3 ⊢ (𝐴 ∈ V → (Ord suc 𝐴 → Ord 𝐴)) |
10 | 5, 9 | impbid 211 | . 2 ⊢ (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴)) |
11 | sucprc 6398 | . . . 4 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) | |
12 | 11 | eqcomd 2743 | . . 3 ⊢ (¬ 𝐴 ∈ V → 𝐴 = suc 𝐴) |
13 | ordeq 6329 | . . 3 ⊢ (𝐴 = suc 𝐴 → (Ord 𝐴 ↔ Ord suc 𝐴)) | |
14 | 12, 13 | syl 17 | . 2 ⊢ (¬ 𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴)) |
15 | 10, 14 | pm2.61i 182 | 1 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3448 Ord word 6321 Oncon0 6322 suc csuc 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-tr 5228 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-ord 6325 df-on 6326 df-suc 6328 |
This theorem is referenced by: (None) |
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