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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ordpss | Structured version Visualization version GIF version | ||
| Description: ordelpss 6412 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.) | 
| Ref | Expression | 
|---|---|
| ordpss | ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ⊊ 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ordelord 6406 | . . . 4 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → Ord 𝐴) | |
| 2 | 1 | ex 412 | . . 3 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → Ord 𝐴)) | 
| 3 | 2 | ancrd 551 | . 2 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → (Ord 𝐴 ∧ 𝐴 ∈ 𝐵))) | 
| 4 | ordelpss 6412 | . . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝐴 ⊊ 𝐵)) | |
| 5 | 4 | ancoms 458 | . . . 4 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴 ∈ 𝐵 ↔ 𝐴 ⊊ 𝐵)) | 
| 6 | 5 | biimpd 229 | . . 3 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴 ∈ 𝐵 → 𝐴 ⊊ 𝐵)) | 
| 7 | 6 | expimpd 453 | . 2 ⊢ (Ord 𝐵 → ((Ord 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊊ 𝐵)) | 
| 8 | 3, 7 | syld 47 | 1 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ⊊ 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ⊊ wpss 3952 Ord word 6383 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 | 
| This theorem is referenced by: (None) | 
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