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Mirrors > Home > MPE Home > Th. List > Mathboxes > ordpss | Structured version Visualization version GIF version |
Description: ordelpss 6413 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 6407 | . . . 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 6413 | . . . . 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 2105 ⊊ wpss 3963 Ord word 6384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-ord 6388 |
This theorem is referenced by: (None) |
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