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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj970 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 34641. 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 34434 | . . . 4 ⊢ (𝜒 → 𝑛 ∈ 𝐷) |
3 | 2 | 3ad2ant1 1131 | . . 3 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑛 ∈ 𝐷) |
4 | 3 | adantl 481 | . 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑛 ∈ 𝐷) |
5 | simpr3 1194 | . 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 34399 | . . . 4 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω) |
8 | peano2 7896 | . . . . 5 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω) | |
9 | eleq1 2817 | . . . . 5 ⊢ (𝑝 = suc 𝑛 → (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω)) | |
10 | bianir 1057 | . . . . 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 579 | . . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ ω) |
13 | df-suc 6375 | . . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛}) | |
14 | 13 | eqeq2i 2741 | . . . . 5 ⊢ (𝑝 = suc 𝑛 ↔ 𝑝 = (𝑛 ∪ {𝑛})) |
15 | ssun2 4173 | . . . . . . 7 ⊢ {𝑛} ⊆ (𝑛 ∪ {𝑛}) | |
16 | vex 3475 | . . . . . . . 8 ⊢ 𝑛 ∈ V | |
17 | 16 | snnz 4781 | . . . . . . 7 ⊢ {𝑛} ≠ ∅ |
18 | ssn0 4401 | . . . . . . 7 ⊢ (({𝑛} ⊆ (𝑛 ∪ {𝑛}) ∧ {𝑛} ≠ ∅) → (𝑛 ∪ {𝑛}) ≠ ∅) | |
19 | 15, 17, 18 | mp2an 691 | . . . . . 6 ⊢ (𝑛 ∪ {𝑛}) ≠ ∅ |
20 | neeq1 3000 | . . . . . 6 ⊢ (𝑝 = (𝑛 ∪ {𝑛}) → (𝑝 ≠ ∅ ↔ (𝑛 ∪ {𝑛}) ≠ ∅)) | |
21 | 19, 20 | mpbiri 258 | . . . . 5 ⊢ (𝑝 = (𝑛 ∪ {𝑛}) → 𝑝 ≠ ∅) |
22 | 14, 21 | sylbi 216 | . . . 4 ⊢ (𝑝 = suc 𝑛 → 𝑝 ≠ ∅) |
23 | 22 | adantl 481 | . . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ≠ ∅) |
24 | 6 | eleq2i 2821 | . . . 4 ⊢ (𝑝 ∈ 𝐷 ↔ 𝑝 ∈ (ω ∖ {∅})) |
25 | eldifsn 4791 | . . . 4 ⊢ (𝑝 ∈ (ω ∖ {∅}) ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅)) | |
26 | 24, 25 | bitri 275 | . . 3 ⊢ (𝑝 ∈ 𝐷 ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅)) |
27 | 12, 23, 26 | sylanbrc 582 | . 2 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ 𝐷) |
28 | 4, 5, 27 | syl2anc 583 | 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 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∖ cdif 3944 ∪ cun 3945 ⊆ wss 3947 ∅c0 4323 {csn 4629 suc csuc 6371 Fn wfn 6543 ωcom 7870 ∧ w-bnj17 34317 FrSe w-bnj15 34323 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-om 7871 df-bnj17 34318 |
This theorem is referenced by: bnj910 34579 |
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