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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 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 |
---|---|
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 33472 | . . . 4 ⊢ (𝜒 → 𝑛 ∈ 𝐷) |
3 | 2 | 3ad2ant1 1134 | . . 3 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑛 ∈ 𝐷) |
4 | 3 | adantl 483 | . 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑛 ∈ 𝐷) |
5 | simpr3 1197 | . 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 33437 | . . . 4 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω) |
8 | peano2 7828 | . . . . 5 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω) | |
9 | eleq1 2822 | . . . . 5 ⊢ (𝑝 = suc 𝑛 → (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω)) | |
10 | bianir 1058 | . . . . 5 ⊢ ((suc 𝑛 ∈ ω ∧ (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω)) → 𝑝 ∈ ω) | |
11 | 8, 9, 10 | syl2an 597 | . . . 4 ⊢ ((𝑛 ∈ ω ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ ω) |
12 | 7, 11 | sylan 581 | . . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ ω) |
13 | df-suc 6324 | . . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛}) | |
14 | 13 | eqeq2i 2746 | . . . . 5 ⊢ (𝑝 = suc 𝑛 ↔ 𝑝 = (𝑛 ∪ {𝑛})) |
15 | ssun2 4134 | . . . . . . 7 ⊢ {𝑛} ⊆ (𝑛 ∪ {𝑛}) | |
16 | vex 3448 | . . . . . . . 8 ⊢ 𝑛 ∈ V | |
17 | 16 | snnz 4738 | . . . . . . 7 ⊢ {𝑛} ≠ ∅ |
18 | ssn0 4361 | . . . . . . 7 ⊢ (({𝑛} ⊆ (𝑛 ∪ {𝑛}) ∧ {𝑛} ≠ ∅) → (𝑛 ∪ {𝑛}) ≠ ∅) | |
19 | 15, 17, 18 | mp2an 691 | . . . . . 6 ⊢ (𝑛 ∪ {𝑛}) ≠ ∅ |
20 | neeq1 3003 | . . . . . 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 483 | . . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ≠ ∅) |
24 | 6 | eleq2i 2826 | . . . 4 ⊢ (𝑝 ∈ 𝐷 ↔ 𝑝 ∈ (ω ∖ {∅})) |
25 | eldifsn 4748 | . . . 4 ⊢ (𝑝 ∈ (ω ∖ {∅}) ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅)) | |
26 | 24, 25 | bitri 275 | . . 3 ⊢ (𝑝 ∈ 𝐷 ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅)) |
27 | 12, 23, 26 | sylanbrc 584 | . 2 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ 𝐷) |
28 | 4, 5, 27 | syl2anc 585 | 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∖ cdif 3908 ∪ cun 3909 ⊆ wss 3911 ∅c0 4283 {csn 4587 suc csuc 6320 Fn wfn 6492 ωcom 7803 ∧ w-bnj17 33355 FrSe w-bnj15 33361 |
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-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
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-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 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-br 5107 df-opab 5169 df-tr 5224 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-om 7804 df-bnj17 33356 |
This theorem is referenced by: bnj910 33617 |
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