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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj970 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32179. 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 |
---|---|
bnj970.3 | ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
bnj970.10 | ⊢ 𝐷 = (ω ∖ {∅}) |
Ref | Expression |
---|---|
bnj970 | ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑝 ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj970.3 | . . . . 5 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
2 | 1 | bnj1232 31974 | . . . 4 ⊢ (𝜒 → 𝑛 ∈ 𝐷) |
3 | 2 | 3ad2ant1 1125 | . . 3 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑛 ∈ 𝐷) |
4 | 3 | adantl 482 | . 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑛 ∈ 𝐷) |
5 | simpr3 1188 | . 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑝 = suc 𝑛) | |
6 | bnj970.10 | . . . . 5 ⊢ 𝐷 = (ω ∖ {∅}) | |
7 | 6 | bnj923 31938 | . . . 4 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω) |
8 | peano2 7591 | . . . . 5 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω) | |
9 | eleq1 2897 | . . . . 5 ⊢ (𝑝 = suc 𝑛 → (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω)) | |
10 | bianir 1050 | . . . . 5 ⊢ ((suc 𝑛 ∈ ω ∧ (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω)) → 𝑝 ∈ ω) | |
11 | 8, 9, 10 | syl2an 595 | . . . 4 ⊢ ((𝑛 ∈ ω ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ ω) |
12 | 7, 11 | sylan 580 | . . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ ω) |
13 | df-suc 6190 | . . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛}) | |
14 | 13 | eqeq2i 2831 | . . . . 5 ⊢ (𝑝 = suc 𝑛 ↔ 𝑝 = (𝑛 ∪ {𝑛})) |
15 | ssun2 4146 | . . . . . . 7 ⊢ {𝑛} ⊆ (𝑛 ∪ {𝑛}) | |
16 | vex 3495 | . . . . . . . 8 ⊢ 𝑛 ∈ V | |
17 | 16 | snnz 4703 | . . . . . . 7 ⊢ {𝑛} ≠ ∅ |
18 | ssn0 4351 | . . . . . . 7 ⊢ (({𝑛} ⊆ (𝑛 ∪ {𝑛}) ∧ {𝑛} ≠ ∅) → (𝑛 ∪ {𝑛}) ≠ ∅) | |
19 | 15, 17, 18 | mp2an 688 | . . . . . 6 ⊢ (𝑛 ∪ {𝑛}) ≠ ∅ |
20 | neeq1 3075 | . . . . . 6 ⊢ (𝑝 = (𝑛 ∪ {𝑛}) → (𝑝 ≠ ∅ ↔ (𝑛 ∪ {𝑛}) ≠ ∅)) | |
21 | 19, 20 | mpbiri 259 | . . . . 5 ⊢ (𝑝 = (𝑛 ∪ {𝑛}) → 𝑝 ≠ ∅) |
22 | 14, 21 | sylbi 218 | . . . 4 ⊢ (𝑝 = suc 𝑛 → 𝑝 ≠ ∅) |
23 | 22 | adantl 482 | . . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ≠ ∅) |
24 | 6 | eleq2i 2901 | . . . 4 ⊢ (𝑝 ∈ 𝐷 ↔ 𝑝 ∈ (ω ∖ {∅})) |
25 | eldifsn 4711 | . . . 4 ⊢ (𝑝 ∈ (ω ∖ {∅}) ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅)) | |
26 | 24, 25 | bitri 276 | . . 3 ⊢ (𝑝 ∈ 𝐷 ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅)) |
27 | 12, 23, 26 | sylanbrc 583 | . 2 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ 𝐷) |
28 | 4, 5, 27 | syl2anc 584 | 1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑝 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∖ cdif 3930 ∪ cun 3931 ⊆ wss 3933 ∅c0 4288 {csn 4557 suc csuc 6186 Fn wfn 6343 ωcom 7569 ∧ w-bnj17 31855 FrSe w-bnj15 31861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-om 7570 df-bnj17 31856 |
This theorem is referenced by: bnj910 32119 |
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