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Mirrors > Home > MPE Home > Th. List > Mathboxes > ordpss | Structured version Visualization version GIF version |
Description: ordelpss 6397 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 6391 | . . . 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 6397 | . . . . 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 228 | . . 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 205 ∧ wa 395 ∈ wcel 2099 ⊊ wpss 3948 Ord word 6368 |
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 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
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 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-ord 6372 |
This theorem is referenced by: (None) |
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