![]() |
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 33679. 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 33362 | . . . 4 ⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴)) | |
3 | 2 | simplbi 499 | . . 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 33663 | . . . . 5 ⊢ trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴 | |
7 | ssinss1 4198 | . . . . 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 3979 | . . 3 ⊢ 𝐶 ⊆ 𝐴 |
10 | 9 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐴) |
11 | bnj1177.17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐴) | |
12 | bnj906 33599 | . . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) | |
13 | 1, 11, 12 | syl2anc 585 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
14 | 13 | ssrind 4196 | . . . . 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 1147 | . . . . . . . . 9 ⊢ (𝜓 → 𝑦 ∈ 𝐵) |
18 | 17 | adantl 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵) |
19 | 15, 18 | sseldd 3946 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐴) |
20 | 16 | simp3bi 1148 | . . . . . . . 8 ⊢ (𝜓 → 𝑦𝑅𝑋) |
21 | 20 | adantl 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑦𝑅𝑋) |
22 | bnj1152 33667 | . . . . . . 7 ⊢ (𝑦 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)) | |
23 | 19, 21, 22 | sylanbrc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ pred(𝑋, 𝐴, 𝑅)) |
24 | 23, 18 | elind 4155 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵)) |
25 | 14, 24 | sseldd 3946 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)) |
26 | 25 | ne0d 4296 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅) |
27 | 5 | neeq1i 3005 | . . 3 ⊢ (𝐶 ≠ ∅ ↔ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅) |
28 | 26, 27 | sylibr 233 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≠ ∅) |
29 | bnj893 33597 | . . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V) | |
30 | 1, 11, 29 | syl2anc 585 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → trCl(𝑋, 𝐴, 𝑅) ∈ V) |
31 | inex1g 5277 | . . . 4 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∈ V → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ∈ V) | |
32 | 5, 31 | eqeltrid 2838 | . . 3 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∈ V → 𝐶 ∈ V) |
33 | 30, 32 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ V) |
34 | 4, 10, 28, 33 | bnj951 33444 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝑅 Fr 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ ∅ ∧ 𝐶 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 Vcvv 3444 ∩ cin 3910 ⊆ wss 3911 ∅c0 4283 class class class wbr 5106 Fr wfr 5586 ∧ w-bnj17 33355 predc-bnj14 33357 Se w-bnj13 33359 FrSe w-bnj15 33361 trClc-bnj18 33363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-reg 9533 ax-inf2 9582 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-om 7804 df-1o 8413 df-bnj17 33356 df-bnj14 33358 df-bnj13 33360 df-bnj15 33362 df-bnj18 33364 |
This theorem is referenced by: bnj1190 33677 |
Copyright terms: Public domain | W3C validator |