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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1175 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 35003. 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 34698 | . . . . 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 34680 | . . . . 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 299 | . . . 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 624 | . . . 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 278 | . . 3 ⊢ (𝜒 ↔ (𝜃 ∧ 𝑤𝑅𝑧)) |
8 | bnj1125 34985 | . . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) | |
9 | 1, 8 | bnj835 34752 | . . . 4 ⊢ (𝜒 → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
10 | bnj906 34923 | . . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅)) | |
11 | 1, 10 | bnj836 34753 | . . . . 5 ⊢ (𝜒 → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅)) |
12 | bnj1152 34991 | . . . . . . 7 ⊢ (𝑤 ∈ pred(𝑧, 𝐴, 𝑅) ↔ (𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧)) | |
13 | 12 | biimpri 228 | . . . . . 6 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧) → 𝑤 ∈ pred(𝑧, 𝐴, 𝑅)) |
14 | 1, 13 | bnj837 34754 | . . . . 5 ⊢ (𝜒 → 𝑤 ∈ pred(𝑧, 𝐴, 𝑅)) |
15 | 11, 14 | sseldd 3996 | . . . 4 ⊢ (𝜒 → 𝑤 ∈ trCl(𝑧, 𝐴, 𝑅)) |
16 | 9, 15 | sseldd 3996 | . . 3 ⊢ (𝜒 → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)) |
17 | 7, 16 | sylbir 235 | . 2 ⊢ ((𝜃 ∧ 𝑤𝑅𝑧) → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)) |
18 | 17 | ex 412 | 1 ⊢ (𝜃 → (𝑤𝑅𝑧 → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 class class class wbr 5148 ∧ w-bnj17 34679 predc-bnj14 34681 FrSe w-bnj15 34685 trClc-bnj18 34687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-reg 9630 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1o 8505 df-bnj17 34680 df-bnj14 34682 df-bnj13 34684 df-bnj15 34686 df-bnj18 34688 df-bnj19 34690 |
This theorem is referenced by: bnj1190 35001 |
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