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Mirrors > Home > MPE Home > Th. List > onmindif | Structured version Visualization version GIF version |
Description: When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass. (Contributed by NM, 1-Dec-2003.) |
Ref | Expression |
---|---|
onmindif | ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → 𝐵 ∈ ∩ (𝐴 ∖ suc 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3726 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ suc 𝐵)) | |
2 | ssel2 3740 | . . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On) | |
3 | ontri1 5919 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝑥)) | |
4 | onsssuc 5975 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ⊆ 𝐵 ↔ 𝑥 ∈ suc 𝐵)) | |
5 | 3, 4 | bitr3d 270 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵 ∈ 𝑥 ↔ 𝑥 ∈ suc 𝐵)) |
6 | 5 | con1bid 344 | . . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝑥 ∈ suc 𝐵 ↔ 𝐵 ∈ 𝑥)) |
7 | 2, 6 | sylan 489 | . . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝐵 ∈ On) → (¬ 𝑥 ∈ suc 𝐵 ↔ 𝐵 ∈ 𝑥)) |
8 | 7 | biimpd 219 | . . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝐵 ∈ On) → (¬ 𝑥 ∈ suc 𝐵 → 𝐵 ∈ 𝑥)) |
9 | 8 | exp31 631 | . . . . . 6 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → (𝐵 ∈ On → (¬ 𝑥 ∈ suc 𝐵 → 𝐵 ∈ 𝑥)))) |
10 | 9 | com23 86 | . . . . 5 ⊢ (𝐴 ⊆ On → (𝐵 ∈ On → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ suc 𝐵 → 𝐵 ∈ 𝑥)))) |
11 | 10 | imp4b 614 | . . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ suc 𝐵) → 𝐵 ∈ 𝑥)) |
12 | 1, 11 | syl5bi 232 | . . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → (𝑥 ∈ (𝐴 ∖ suc 𝐵) → 𝐵 ∈ 𝑥)) |
13 | 12 | ralrimiv 3104 | . 2 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵 ∈ 𝑥) |
14 | elintg 4636 | . . 3 ⊢ (𝐵 ∈ On → (𝐵 ∈ ∩ (𝐴 ∖ suc 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵 ∈ 𝑥)) | |
15 | 14 | adantl 473 | . 2 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → (𝐵 ∈ ∩ (𝐴 ∖ suc 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵 ∈ 𝑥)) |
16 | 13, 15 | mpbird 247 | 1 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → 𝐵 ∈ ∩ (𝐴 ∖ suc 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 ∈ wcel 2140 ∀wral 3051 ∖ cdif 3713 ⊆ wss 3716 ∩ cint 4628 Oncon0 5885 suc csuc 5887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-sep 4934 ax-nul 4942 ax-pr 5056 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3343 df-sbc 3578 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-sn 4323 df-pr 4325 df-op 4329 df-uni 4590 df-int 4629 df-br 4806 df-opab 4866 df-tr 4906 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-we 5228 df-ord 5888 df-on 5889 df-suc 5891 |
This theorem is referenced by: unblem3 8382 fin23lem26 9360 |
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