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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 32392. 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 32109 | . . . . . 6 ⊢ (𝑛 ∈ 𝐷 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚) |
4 | 1, 3 | bnj769 32143 | . . . . 5 ⊢ (𝜒 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚) |
5 | 4 | bnj1196 32176 | . . . 4 ⊢ (𝜒 → ∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚)) |
6 | vex 3444 | . . . . . 6 ⊢ 𝑛 ∈ V | |
7 | 6 | sucex 7506 | . . . . 5 ⊢ suc 𝑛 ∈ V |
8 | 7 | isseti 3455 | . . . 4 ⊢ ∃𝑝 𝑝 = suc 𝑛 |
9 | 5, 8 | jctir 524 | . . 3 ⊢ (𝜒 → (∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛)) |
10 | exdistr 1955 | . . . 4 ⊢ (∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛) ↔ ∃𝑚((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛)) | |
11 | 19.41v 1950 | . . . 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 279 | . . 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 221 | . 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 1086 | . . . 4 ⊢ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛)) | |
16 | 14, 15 | bitri 278 | . . 3 ⊢ (𝜏 ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛)) |
17 | 16 | 2exbii 1850 | . 2 ⊢ (∃𝑚∃𝑝𝜏 ↔ ∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛)) |
18 | 13, 17 | sylibr 237 | 1 ⊢ (𝜒 → ∃𝑚∃𝑝𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∃wrex 3107 ∖ cdif 3878 ∅c0 4243 {csn 4525 suc csuc 6161 Fn wfn 6319 ωcom 7560 ∧ w-bnj17 32066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-om 7561 df-bnj17 32067 |
This theorem is referenced by: bnj996 32338 |
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