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Mirrors > Home > MPE Home > Th. List > ordnbtwn | Structured version Visualization version GIF version |
Description: There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 21-Jun-1998.) (Proof shortened by JJ, 24-Sep-2021.) |
Ref | Expression |
---|---|
ordnbtwn | ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordirr 6202 | . . 3 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
2 | ordn2lp 6204 | . . . 4 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) | |
3 | pm2.24 124 | . . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐴 ∈ 𝐴)) | |
4 | eleq2 2899 | . . . . . . 7 ⊢ (𝐵 = 𝐴 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐴)) | |
5 | 4 | biimpac 481 | . . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 = 𝐴) → 𝐴 ∈ 𝐴) |
6 | 5 | a1d 25 | . . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 = 𝐴) → (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐴 ∈ 𝐴)) |
7 | 3, 6 | jaodan 954 | . . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) → (¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐴 ∈ 𝐴)) |
8 | 2, 7 | syl5com 31 | . . 3 ⊢ (Ord 𝐴 → ((𝐴 ∈ 𝐵 ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) → 𝐴 ∈ 𝐴)) |
9 | 1, 8 | mtod 200 | . 2 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))) |
10 | elsuci 6250 | . . 3 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) | |
11 | 10 | anim2i 618 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴) → (𝐴 ∈ 𝐵 ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))) |
12 | 9, 11 | nsyl 142 | 1 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1531 ∈ wcel 2108 Ord word 6183 suc csuc 6186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-fr 5507 df-we 5509 df-ord 6187 df-suc 6190 |
This theorem is referenced by: onnbtwn 6275 ordsucss 7525 |
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