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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1177 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 34674. 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 34357 | . . . 4 ⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴)) | |
3 | 2 | simplbi 496 | . . 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 34658 | . . . . 5 ⊢ trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴 | |
7 | ssinss1 4240 | . . . . 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 4016 | . . 3 ⊢ 𝐶 ⊆ 𝐴 |
10 | 9 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐴) |
11 | bnj1177.17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐴) | |
12 | bnj906 34594 | . . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) | |
13 | 1, 11, 12 | syl2anc 582 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
14 | 13 | ssrind 4238 | . . . . 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 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵) |
19 | 15, 18 | sseldd 3983 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐴) |
20 | 16 | simp3bi 1144 | . . . . . . . 8 ⊢ (𝜓 → 𝑦𝑅𝑋) |
21 | 20 | adantl 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑦𝑅𝑋) |
22 | bnj1152 34662 | . . . . . . 7 ⊢ (𝑦 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋)) | |
23 | 19, 21, 22 | sylanbrc 581 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ pred(𝑋, 𝐴, 𝑅)) |
24 | 23, 18 | elind 4196 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵)) |
25 | 14, 24 | sseldd 3983 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)) |
26 | 25 | ne0d 4339 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅) |
27 | 5 | neeq1i 3002 | . . 3 ⊢ (𝐶 ≠ ∅ ↔ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅) |
28 | 26, 27 | sylibr 233 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≠ ∅) |
29 | bnj893 34592 | . . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V) | |
30 | 1, 11, 29 | syl2anc 582 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → trCl(𝑋, 𝐴, 𝑅) ∈ V) |
31 | inex1g 5323 | . . . 4 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∈ V → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ∈ V) | |
32 | 5, 31 | eqeltrid 2833 | . . 3 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∈ V → 𝐶 ∈ V) |
33 | 30, 32 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ V) |
34 | 4, 10, 28, 33 | bnj951 34439 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝑅 Fr 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ ∅ ∧ 𝐶 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 Vcvv 3473 ∩ cin 3948 ⊆ wss 3949 ∅c0 4326 class class class wbr 5152 Fr wfr 5634 ∧ w-bnj17 34350 predc-bnj14 34352 Se w-bnj13 34354 FrSe w-bnj15 34356 trClc-bnj18 34358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-reg 9623 ax-inf2 9672 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-om 7877 df-1o 8493 df-bnj17 34351 df-bnj14 34353 df-bnj13 34355 df-bnj15 34357 df-bnj18 34359 |
This theorem is referenced by: bnj1190 34672 |
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