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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1175 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 31078. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1175.3 | ⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) |
bnj1175.4 | ⊢ (𝜒 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧))) |
bnj1175.5 | ⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴)) |
Ref | Expression |
---|---|
bnj1175 | ⊢ (𝜃 → (𝑤𝑅𝑧 → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1175.4 | . . . . 5 ⊢ (𝜒 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧))) | |
2 | bnj255 30771 | . . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧))) | |
3 | df-bnj17 30753 | . . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧) ↔ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) ∧ 𝑤𝑅𝑧)) | |
4 | 1, 2, 3 | 3bitr2i 288 | . . . 4 ⊢ (𝜒 ↔ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) ∧ 𝑤𝑅𝑧)) |
5 | bnj1175.5 | . . . . 5 ⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴)) | |
6 | 5 | anbi1i 731 | . . . 4 ⊢ ((𝜃 ∧ 𝑤𝑅𝑧) ↔ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) ∧ 𝑤𝑅𝑧)) |
7 | 4, 6 | bitr4i 267 | . . 3 ⊢ (𝜒 ↔ (𝜃 ∧ 𝑤𝑅𝑧)) |
8 | bnj1125 31060 | . . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) | |
9 | 1, 8 | bnj835 30829 | . . . 4 ⊢ (𝜒 → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
10 | bnj906 31000 | . . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅)) | |
11 | 1, 10 | bnj836 30830 | . . . . 5 ⊢ (𝜒 → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅)) |
12 | bnj1152 31066 | . . . . . . 7 ⊢ (𝑤 ∈ pred(𝑧, 𝐴, 𝑅) ↔ (𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧)) | |
13 | 12 | biimpri 218 | . . . . . 6 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧) → 𝑤 ∈ pred(𝑧, 𝐴, 𝑅)) |
14 | 1, 13 | bnj837 30831 | . . . . 5 ⊢ (𝜒 → 𝑤 ∈ pred(𝑧, 𝐴, 𝑅)) |
15 | 11, 14 | sseldd 3604 | . . . 4 ⊢ (𝜒 → 𝑤 ∈ trCl(𝑧, 𝐴, 𝑅)) |
16 | 9, 15 | sseldd 3604 | . . 3 ⊢ (𝜒 → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)) |
17 | 7, 16 | sylbir 225 | . 2 ⊢ ((𝜃 ∧ 𝑤𝑅𝑧) → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)) |
18 | 17 | ex 450 | 1 ⊢ (𝜃 → (𝑤𝑅𝑧 → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 class class class wbr 4653 ∧ w-bnj17 30752 predc-bnj14 30754 FrSe w-bnj15 30758 trClc-bnj18 30760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-reg 8497 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-1o 7560 df-bnj17 30753 df-bnj14 30755 df-bnj13 30757 df-bnj15 30759 df-bnj18 30761 df-bnj19 30763 |
This theorem is referenced by: bnj1190 31076 |
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