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Mirrors > Home > MPE Home > Th. List > Mathboxes > ordeldif | Structured version Visualization version GIF version |
Description: Membership in the difference of ordinals. (Contributed by RP, 15-Jan-2025.) |
Ref | Expression |
---|---|
ordeldif | ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐶 ∈ (𝐴 ∖ 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3954 | . 2 ⊢ (𝐶 ∈ (𝐴 ∖ 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐵)) | |
2 | simpr 484 | . . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord 𝐵) | |
3 | ordelord 6385 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐶 ∈ 𝐴) → Ord 𝐶) | |
4 | 3 | adantlr 714 | . . . . 5 ⊢ (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝐶 ∈ 𝐴) → Ord 𝐶) |
5 | ordtri1 6396 | . . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 ⊆ 𝐶 ↔ ¬ 𝐶 ∈ 𝐵)) | |
6 | 2, 4, 5 | syl2an2r 684 | . . . 4 ⊢ (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐵 ⊆ 𝐶 ↔ ¬ 𝐶 ∈ 𝐵)) |
7 | 6 | bicomd 222 | . . 3 ⊢ (((Ord 𝐴 ∧ Ord 𝐵) ∧ 𝐶 ∈ 𝐴) → (¬ 𝐶 ∈ 𝐵 ↔ 𝐵 ⊆ 𝐶)) |
8 | 7 | pm5.32da 578 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶))) |
9 | 1, 8 | bitrid 283 | 1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐶 ∈ (𝐴 ∖ 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2099 ∖ cdif 3941 ⊆ wss 3944 Ord word 6362 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-tr 5260 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-ord 6366 |
This theorem is referenced by: tfsconcatlem 42737 tfsconcatfv2 42741 tfsconcatrn 42743 tfsconcatb0 42745 tfsconcatrev 42749 |
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