Mathbox for Andrew Salmon |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ordpss | Structured version Visualization version GIF version |
Description: ordelpss 5668 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 5662 | . . . 4 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → Ord 𝐴) | |
2 | 1 | ex 449 | . . 3 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → Ord 𝐴)) |
3 | 2 | ancrd 575 | . 2 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → (Ord 𝐴 ∧ 𝐴 ∈ 𝐵))) |
4 | ordelpss 5668 | . . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝐴 ⊊ 𝐵)) | |
5 | 4 | ancoms 468 | . . . 4 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴 ∈ 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
6 | 5 | biimpd 218 | . . 3 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴 ∈ 𝐵 → 𝐴 ⊊ 𝐵)) |
7 | 6 | expimpd 627 | . 2 ⊢ (Ord 𝐵 → ((Ord 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊊ 𝐵)) |
8 | 3, 7 | syld 46 | 1 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ⊊ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ⊊ wpss 3541 Ord word 5639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |