![]() |
Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1177 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32392. 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 |
---|---|
bnj1177.2 | ⊢ (𝜓 ↔ (𝑋 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝑅𝑋)) |
bnj1177.3 | ⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) |
bnj1177.9 | ⊢ ((𝜑 ∧ 𝜓) → 𝑅 FrSe 𝐴) |
bnj1177.13 | ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ⊆ 𝐴) |
bnj1177.17 | ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐴) |
Ref | Expression |
---|---|
bnj1177 | ⊢ ((𝜑 ∧ 𝜓) → (𝑅 Fr 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ ∅ ∧ 𝐶 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1177.9 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 FrSe 𝐴) | |
2 | df-bnj15 32073 | . . . 4 ⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴)) | |
3 | 2 | simplbi 501 | . . 3 ⊢ (𝑅 FrSe 𝐴 → 𝑅 Fr 𝐴) |
4 | 1, 3 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 Fr 𝐴) |
5 | bnj1177.3 | . . . 4 ⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) | |
6 | bnj1147 32376 | . . . . 5 ⊢ trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴 | |
7 | ssinss1 4164 | . . . . 5 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴 → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ⊆ 𝐴) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ⊆ 𝐴 |
9 | 5, 8 | eqsstri 3949 | . . 3 ⊢ 𝐶 ⊆ 𝐴 |
10 | 9 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐴) |
11 | bnj1177.17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐴) | |
12 | bnj906 32312 | . . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) | |
13 | 1, 11, 12 | syl2anc 587 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
14 | 13 | ssrind 4162 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵) ⊆ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)) |
15 | bnj1177.13 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ⊆ 𝐴) | |
16 | bnj1177.2 | . . . . . . . . . 10 ⊢ (𝜓 ↔ (𝑋 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝑅𝑋)) | |
17 | 16 | simp2bi 1143 | . . . . . . . . 9 ⊢ (𝜓 → 𝑦 ∈ 𝐵) |
18 | 17 | adantl 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵) |
19 | 15, 18 | sseldd 3916 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐴) |
20 | 16 | simp3bi 1144 | . . . . . . . 8 ⊢ (𝜓 → 𝑦𝑅𝑋) |
21 | 20 | adantl 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑦𝑅𝑋) |
22 | bnj1152 32380 | . . . . . . 7 ⊢ (𝑦 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)) | |
23 | 19, 21, 22 | sylanbrc 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ pred(𝑋, 𝐴, 𝑅)) |
24 | 23, 18 | elind 4121 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵)) |
25 | 14, 24 | sseldd 3916 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)) |
26 | 25 | ne0d 4251 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅) |
27 | 5 | neeq1i 3051 | . . 3 ⊢ (𝐶 ≠ ∅ ↔ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅) |
28 | 26, 27 | sylibr 237 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≠ ∅) |
29 | bnj893 32310 | . . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V) | |
30 | 1, 11, 29 | syl2anc 587 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → trCl(𝑋, 𝐴, 𝑅) ∈ V) |
31 | inex1g 5187 | . . . 4 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∈ V → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ∈ V) | |
32 | 5, 31 | eqeltrid 2894 | . . 3 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∈ V → 𝐶 ∈ V) |
33 | 30, 32 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ V) |
34 | 4, 10, 28, 33 | bnj951 32157 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝑅 Fr 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ ∅ ∧ 𝐶 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 Fr wfr 5475 ∧ w-bnj17 32066 predc-bnj14 32068 Se w-bnj13 32070 FrSe w-bnj15 32072 trClc-bnj18 32074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-1o 8085 df-bnj17 32067 df-bnj14 32069 df-bnj13 32071 df-bnj15 32073 df-bnj18 32075 |
This theorem is referenced by: bnj1190 32390 |
Copyright terms: Public domain | W3C validator |