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Mirrors > Home > MPE Home > Th. List > Mathboxes > ordpss | Structured version Visualization version GIF version |
Description: ordelpss 6219 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 6213 | . . . 4 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → Ord 𝐴) | |
2 | 1 | ex 415 | . . 3 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → Ord 𝐴)) |
3 | 2 | ancrd 554 | . 2 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → (Ord 𝐴 ∧ 𝐴 ∈ 𝐵))) |
4 | ordelpss 6219 | . . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝐴 ⊊ 𝐵)) | |
5 | 4 | ancoms 461 | . . . 4 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴 ∈ 𝐵 ↔ 𝐴 ⊊ 𝐵)) |
6 | 5 | biimpd 231 | . . 3 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴 ∈ 𝐵 → 𝐴 ⊊ 𝐵)) |
7 | 6 | expimpd 456 | . 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 208 ∧ wa 398 ∈ wcel 2114 ⊊ wpss 3937 Ord word 6190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-ord 6194 |
This theorem is referenced by: (None) |
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