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Mirrors > Home > MPE Home > Th. List > ordsssuc2 | Structured version Visualization version GIF version |
Description: An ordinal subset of an ordinal number belongs to its successor. (Contributed by NM, 1-Feb-2005.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
ordsssuc2 | ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elong 5648 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴)) | |
2 | 1 | biimprd 237 | . . . 4 ⊢ (𝐴 ∈ V → (Ord 𝐴 → 𝐴 ∈ On)) |
3 | 2 | anim1d 586 | . . 3 ⊢ (𝐴 ∈ V → ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On))) |
4 | onsssuc 5730 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) | |
5 | 3, 4 | syl6 34 | . 2 ⊢ (𝐴 ∈ V → ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))) |
6 | annim 440 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) ↔ ¬ (𝐵 ∈ On → 𝐴 ∈ V)) | |
7 | ssexg 4732 | . . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ On) → 𝐴 ∈ V) | |
8 | 7 | ex 449 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ On → 𝐴 ∈ V)) |
9 | elex 3185 | . . . . . . 7 ⊢ (𝐴 ∈ suc 𝐵 → 𝐴 ∈ V) | |
10 | 9 | a1d 25 | . . . . . 6 ⊢ (𝐴 ∈ suc 𝐵 → (𝐵 ∈ On → 𝐴 ∈ V)) |
11 | 8, 10 | pm5.21ni 366 | . . . . 5 ⊢ (¬ (𝐵 ∈ On → 𝐴 ∈ V) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) |
12 | 6, 11 | sylbi 206 | . . . 4 ⊢ ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) |
13 | 12 | expcom 450 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐵 ∈ On → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))) |
14 | 13 | adantld 482 | . 2 ⊢ (¬ 𝐴 ∈ V → ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))) |
15 | 5, 14 | pm2.61i 175 | 1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 Ord word 5639 Oncon0 5640 suc csuc 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-suc 5646 |
This theorem is referenced by: ordunisuc2 6936 |
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