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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1177 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32275. 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 31956 | . . . 4 ⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴)) | |
3 | 2 | simplbi 500 | . . 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 32259 | . . . . 5 ⊢ trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴 | |
7 | ssinss1 4212 | . . . . 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 3999 | . . 3 ⊢ 𝐶 ⊆ 𝐴 |
10 | 9 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐴) |
11 | bnj1177.17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐴) | |
12 | bnj906 32195 | . . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) | |
13 | 1, 11, 12 | syl2anc 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
14 | 13 | ssrind 4210 | . . . . 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 1141 | . . . . . . . . 9 ⊢ (𝜓 → 𝑦 ∈ 𝐵) |
18 | 17 | adantl 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵) |
19 | 15, 18 | sseldd 3966 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐴) |
20 | 16 | simp3bi 1142 | . . . . . . . 8 ⊢ (𝜓 → 𝑦𝑅𝑋) |
21 | 20 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑦𝑅𝑋) |
22 | bnj1152 32263 | . . . . . . 7 ⊢ (𝑦 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)) | |
23 | 19, 21, 22 | sylanbrc 585 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ pred(𝑋, 𝐴, 𝑅)) |
24 | 23, 18 | elind 4169 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵)) |
25 | 14, 24 | sseldd 3966 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)) |
26 | 25 | ne0d 4299 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅) |
27 | 5 | neeq1i 3078 | . . 3 ⊢ (𝐶 ≠ ∅ ↔ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅) |
28 | 26, 27 | sylibr 236 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≠ ∅) |
29 | bnj893 32193 | . . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V) | |
30 | 1, 11, 29 | syl2anc 586 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → trCl(𝑋, 𝐴, 𝑅) ∈ V) |
31 | inex1g 5214 | . . . 4 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∈ V → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ∈ V) | |
32 | 5, 31 | eqeltrid 2915 | . . 3 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∈ V → 𝐶 ∈ V) |
33 | 30, 32 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ V) |
34 | 4, 10, 28, 33 | bnj951 32040 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝑅 Fr 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ ∅ ∧ 𝐶 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 Vcvv 3493 ∩ cin 3933 ⊆ wss 3934 ∅c0 4289 class class class wbr 5057 Fr wfr 5504 ∧ w-bnj17 31949 predc-bnj14 31951 Se w-bnj13 31953 FrSe w-bnj15 31955 trClc-bnj18 31957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-reg 9048 ax-inf2 9096 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7573 df-1o 8094 df-bnj17 31950 df-bnj14 31952 df-bnj13 31954 df-bnj15 31956 df-bnj18 31958 |
This theorem is referenced by: bnj1190 32273 |
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