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Mirrors > Home > MPE Home > Th. List > ordsucOLD | Structured version Visualization version GIF version |
Description: Obsolete version of ordsuc 7814 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 6372 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴)) | |
2 | onsuc 7812 | . . . . 5 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
3 | eloni 6374 | . . . . 5 ⊢ (suc 𝐴 ∈ On → Ord suc 𝐴) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐴 ∈ On → Ord suc 𝐴) |
5 | 1, 4 | syl6bir 253 | . . 3 ⊢ (𝐴 ∈ V → (Ord 𝐴 → Ord suc 𝐴)) |
6 | sucidg 6445 | . . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴) | |
7 | ordelord 6386 | . . . . 5 ⊢ ((Ord suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → Ord 𝐴) | |
8 | 7 | ex 411 | . . . 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 6440 | . . . 4 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴) | |
12 | 11 | eqcomd 2731 | . . 3 ⊢ (¬ 𝐴 ∈ V → 𝐴 = suc 𝐴) |
13 | ordeq 6371 | . . 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 1533 ∈ wcel 2098 Vcvv 3463 Ord word 6363 Oncon0 6364 suc csuc 6366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-tr 5261 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-ord 6367 df-on 6368 df-suc 6370 |
This theorem is referenced by: (None) |
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