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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj986 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 33959. 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 |
---|---|
bnj986.3 | ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
bnj986.10 | ⊢ 𝐷 = (ω ∖ {∅}) |
bnj986.15 | ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) |
Ref | Expression |
---|---|
bnj986 | ⊢ (𝜒 → ∃𝑚∃𝑝𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj986.3 | . . . . . 6 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
2 | bnj986.10 | . . . . . . 7 ⊢ 𝐷 = (ω ∖ {∅}) | |
3 | 2 | bnj158 33678 | . . . . . 6 ⊢ (𝑛 ∈ 𝐷 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚) |
4 | 1, 3 | bnj769 33711 | . . . . 5 ⊢ (𝜒 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚) |
5 | 4 | bnj1196 33743 | . . . 4 ⊢ (𝜒 → ∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚)) |
6 | vex 3479 | . . . . . 6 ⊢ 𝑛 ∈ V | |
7 | 6 | sucex 7789 | . . . . 5 ⊢ suc 𝑛 ∈ V |
8 | 7 | isseti 3490 | . . . 4 ⊢ ∃𝑝 𝑝 = suc 𝑛 |
9 | 5, 8 | jctir 522 | . . 3 ⊢ (𝜒 → (∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛)) |
10 | exdistr 1959 | . . . 4 ⊢ (∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛) ↔ ∃𝑚((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛)) | |
11 | 19.41v 1954 | . . . 4 ⊢ (∃𝑚((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛) ↔ (∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛)) | |
12 | 10, 11 | bitr2i 276 | . . 3 ⊢ ((∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛) ↔ ∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛)) |
13 | 9, 12 | sylib 217 | . 2 ⊢ (𝜒 → ∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛)) |
14 | bnj986.15 | . . . 4 ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) | |
15 | df-3an 1090 | . . . 4 ⊢ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛)) | |
16 | 14, 15 | bitri 275 | . . 3 ⊢ (𝜏 ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛)) |
17 | 16 | 2exbii 1852 | . 2 ⊢ (∃𝑚∃𝑝𝜏 ↔ ∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛)) |
18 | 13, 17 | sylibr 233 | 1 ⊢ (𝜒 → ∃𝑚∃𝑝𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∃wrex 3071 ∖ cdif 3944 ∅c0 4321 {csn 4627 suc csuc 6363 Fn wfn 6535 ωcom 7850 ∧ w-bnj17 33635 |
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 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 |
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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-om 7851 df-bnj17 33636 |
This theorem is referenced by: bnj996 33905 |
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