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Theorem exmidonfinlem 7049
 Description: Lemma for exmidonfin 7050. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
Hypothesis
Ref Expression
exmidonfinlem.a 𝐴 = {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}
Assertion
Ref Expression
exmidonfinlem (ω = (On ∩ Fin) → DECID 𝜑)
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem exmidonfinlem
Dummy variables 𝑟 𝑠 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpri 3550 . . . . . . . . . 10 (𝑟 ∈ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} → (𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
2 exmidonfinlem.a . . . . . . . . . 10 𝐴 = {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}
31, 2eleq2s 2234 . . . . . . . . 9 (𝑟𝐴 → (𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
4 eleq2 2203 . . . . . . . . . . . 12 (𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} → (𝑠𝑟𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
54biimpcd 158 . . . . . . . . . . 11 (𝑠𝑟 → (𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
6 elrabi 2837 . . . . . . . . . . . . . 14 (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 ∈ {∅})
7 velsn 3544 . . . . . . . . . . . . . 14 (𝑠 ∈ {∅} ↔ 𝑠 = ∅)
86, 7sylib 121 . . . . . . . . . . . . 13 (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 = ∅)
9 biidd 171 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑠 → (𝜑𝜑))
109elrab 2840 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ (𝑠 ∈ {∅} ∧ 𝜑))
1110simprbi 273 . . . . . . . . . . . . . . . 16 (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝜑)
1211notnotd 619 . . . . . . . . . . . . . . 15 (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → ¬ ¬ 𝜑)
13 0ex 4055 . . . . . . . . . . . . . . . . 17 ∅ ∈ V
1413snm 3643 . . . . . . . . . . . . . . . 16 𝑤 𝑤 ∈ {∅}
15 r19.3rmv 3453 . . . . . . . . . . . . . . . 16 (∃𝑤 𝑤 ∈ {∅} → (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑))
1614, 15ax-mp 5 . . . . . . . . . . . . . . 15 (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
1712, 16sylib 121 . . . . . . . . . . . . . 14 (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
18 rabeq0 3392 . . . . . . . . . . . . . 14 ({𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅ ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
1917, 18sylibr 133 . . . . . . . . . . . . 13 (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → {𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅)
208, 19eqtr4d 2175 . . . . . . . . . . . 12 (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠 = {𝑥 ∈ {∅} ∣ ¬ 𝜑})
21 p0ex 4112 . . . . . . . . . . . . . . 15 {∅} ∈ V
2221rabex 4072 . . . . . . . . . . . . . 14 {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ V
2322prid2 3630 . . . . . . . . . . . . 13 {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}
2423, 2eleqtrri 2215 . . . . . . . . . . . 12 {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ 𝐴
2520, 24eqeltrdi 2230 . . . . . . . . . . 11 (𝑠 ∈ {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠𝐴)
265, 25syl6 33 . . . . . . . . . 10 (𝑠𝑟 → (𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} → 𝑠𝐴))
27 eleq2 2203 . . . . . . . . . . . 12 (𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (𝑠𝑟𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
2827biimpcd 158 . . . . . . . . . . 11 (𝑠𝑟 → (𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
29 elrabi 2837 . . . . . . . . . . . . . 14 (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 ∈ {∅})
3029, 7sylib 121 . . . . . . . . . . . . 13 (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 = ∅)
31 biidd 171 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑠 → (¬ 𝜑 ↔ ¬ 𝜑))
3231elrab 2840 . . . . . . . . . . . . . . . 16 (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ↔ (𝑠 ∈ {∅} ∧ ¬ 𝜑))
3332simprbi 273 . . . . . . . . . . . . . . 15 (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ¬ 𝜑)
34 r19.3rmv 3453 . . . . . . . . . . . . . . . 16 (∃𝑤 𝑤 ∈ {∅} → (¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ 𝜑))
3514, 34ax-mp 5 . . . . . . . . . . . . . . 15 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ 𝜑)
3633, 35sylib 121 . . . . . . . . . . . . . 14 (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ∀𝑥 ∈ {∅} ¬ 𝜑)
37 rabeq0 3392 . . . . . . . . . . . . . 14 ({𝑥 ∈ {∅} ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ {∅} ¬ 𝜑)
3836, 37sylibr 133 . . . . . . . . . . . . 13 (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → {𝑥 ∈ {∅} ∣ 𝜑} = ∅)
3930, 38eqtr4d 2175 . . . . . . . . . . . 12 (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠 = {𝑥 ∈ {∅} ∣ 𝜑})
4021rabex 4072 . . . . . . . . . . . . . 14 {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
4140prid1 3629 . . . . . . . . . . . . 13 {𝑥 ∈ {∅} ∣ 𝜑} ∈ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}
4241, 2eleqtrri 2215 . . . . . . . . . . . 12 {𝑥 ∈ {∅} ∣ 𝜑} ∈ 𝐴
4339, 42eqeltrdi 2230 . . . . . . . . . . 11 (𝑠 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠𝐴)
4428, 43syl6 33 . . . . . . . . . 10 (𝑠𝑟 → (𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → 𝑠𝐴))
4526, 44jaod 706 . . . . . . . . 9 (𝑠𝑟 → ((𝑟 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑟 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}) → 𝑠𝐴))
463, 45mpan9 279 . . . . . . . 8 ((𝑟𝐴𝑠𝑟) → 𝑠𝐴)
4746rgen2 2518 . . . . . . 7 𝑟𝐴𝑠𝑟 𝑠𝐴
48 dftr5 4029 . . . . . . 7 (Tr 𝐴 ↔ ∀𝑟𝐴𝑠𝑟 𝑠𝐴)
4947, 48mpbir 145 . . . . . 6 Tr 𝐴
50 elpri 3550 . . . . . . . . 9 (𝑧 ∈ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} → (𝑧 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑧 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
5150, 2eleq2s 2234 . . . . . . . 8 (𝑧𝐴 → (𝑧 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑧 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
52 ordtriexmidlem 4435 . . . . . . . . . . 11 {𝑥 ∈ {∅} ∣ 𝜑} ∈ On
5352ontrci 4349 . . . . . . . . . 10 Tr {𝑥 ∈ {∅} ∣ 𝜑}
54 treq 4032 . . . . . . . . . 10 (𝑧 = {𝑥 ∈ {∅} ∣ 𝜑} → (Tr 𝑧 ↔ Tr {𝑥 ∈ {∅} ∣ 𝜑}))
5553, 54mpbiri 167 . . . . . . . . 9 (𝑧 = {𝑥 ∈ {∅} ∣ 𝜑} → Tr 𝑧)
56 ordtriexmidlem 4435 . . . . . . . . . . 11 {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ On
5756ontrci 4349 . . . . . . . . . 10 Tr {𝑥 ∈ {∅} ∣ ¬ 𝜑}
58 treq 4032 . . . . . . . . . 10 (𝑧 = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (Tr 𝑧 ↔ Tr {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
5957, 58mpbiri 167 . . . . . . . . 9 (𝑧 = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → Tr 𝑧)
6055, 59jaoi 705 . . . . . . . 8 ((𝑧 = {𝑥 ∈ {∅} ∣ 𝜑} ∨ 𝑧 = {𝑥 ∈ {∅} ∣ ¬ 𝜑}) → Tr 𝑧)
6151, 60syl 14 . . . . . . 7 (𝑧𝐴 → Tr 𝑧)
6261rgen 2485 . . . . . 6 𝑧𝐴 Tr 𝑧
63 dford3 4289 . . . . . 6 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑧𝐴 Tr 𝑧))
6449, 62, 63mpbir2an 926 . . . . 5 Ord 𝐴
65 prexg 4133 . . . . . . . 8 (({𝑥 ∈ {∅} ∣ 𝜑} ∈ V ∧ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ V) → {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ∈ V)
6640, 22, 65mp2an 422 . . . . . . 7 {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ∈ V
672, 66eqeltri 2212 . . . . . 6 𝐴 ∈ V
6867elon 4296 . . . . 5 (𝐴 ∈ On ↔ Ord 𝐴)
6964, 68mpbir 145 . . . 4 𝐴 ∈ On
70 2onn 6417 . . . . . 6 2o ∈ ω
71 nnfi 6766 . . . . . 6 (2o ∈ ω → 2o ∈ Fin)
7270, 71ax-mp 5 . . . . 5 2o ∈ Fin
73 pm5.19 695 . . . . . . . . . 10 ¬ (𝜑 ↔ ¬ 𝜑)
7413snm 3643 . . . . . . . . . . 11 𝑦 𝑦 ∈ {∅}
75 r19.3rmv 3453 . . . . . . . . . . 11 (∃𝑦 𝑦 ∈ {∅} → ((𝜑 ↔ ¬ 𝜑) ↔ ∀𝑥 ∈ {∅} (𝜑 ↔ ¬ 𝜑)))
7674, 75ax-mp 5 . . . . . . . . . 10 ((𝜑 ↔ ¬ 𝜑) ↔ ∀𝑥 ∈ {∅} (𝜑 ↔ ¬ 𝜑))
7773, 76mtbi 659 . . . . . . . . 9 ¬ ∀𝑥 ∈ {∅} (𝜑 ↔ ¬ 𝜑)
78 rabbi 2608 . . . . . . . . 9 (∀𝑥 ∈ {∅} (𝜑 ↔ ¬ 𝜑) ↔ {𝑥 ∈ {∅} ∣ 𝜑} = {𝑥 ∈ {∅} ∣ ¬ 𝜑})
7977, 78mtbi 659 . . . . . . . 8 ¬ {𝑥 ∈ {∅} ∣ 𝜑} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}
8079neir 2311 . . . . . . 7 {𝑥 ∈ {∅} ∣ 𝜑} ≠ {𝑥 ∈ {∅} ∣ ¬ 𝜑}
81 pr2ne 7048 . . . . . . . 8 (({𝑥 ∈ {∅} ∣ 𝜑} ∈ V ∧ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ∈ V) → ({{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ≈ 2o ↔ {𝑥 ∈ {∅} ∣ 𝜑} ≠ {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
8240, 22, 81mp2an 422 . . . . . . 7 ({{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ≈ 2o ↔ {𝑥 ∈ {∅} ∣ 𝜑} ≠ {𝑥 ∈ {∅} ∣ ¬ 𝜑})
8380, 82mpbir 145 . . . . . 6 {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} ≈ 2o
842, 83eqbrtri 3949 . . . . 5 𝐴 ≈ 2o
85 enfii 6768 . . . . 5 ((2o ∈ Fin ∧ 𝐴 ≈ 2o) → 𝐴 ∈ Fin)
8672, 84, 85mp2an 422 . . . 4 𝐴 ∈ Fin
8769, 86elini 3260 . . 3 𝐴 ∈ (On ∩ Fin)
88 eleq2 2203 . . 3 (ω = (On ∩ Fin) → (𝐴 ∈ ω ↔ 𝐴 ∈ (On ∩ Fin)))
8987, 88mpbiri 167 . 2 (ω = (On ∩ Fin) → 𝐴 ∈ ω)
90 df1o2 6326 . . . . 5 1o = {∅}
91 1lt2o 6339 . . . . 5 1o ∈ 2o
9290, 91eqeltrri 2213 . . . 4 {∅} ∈ 2o
93 nneneq 6751 . . . . . 6 ((𝐴 ∈ ω ∧ 2o ∈ ω) → (𝐴 ≈ 2o𝐴 = 2o))
9470, 93mpan2 421 . . . . 5 (𝐴 ∈ ω → (𝐴 ≈ 2o𝐴 = 2o))
9584, 94mpbii 147 . . . 4 (𝐴 ∈ ω → 𝐴 = 2o)
9692, 95eleqtrrid 2229 . . 3 (𝐴 ∈ ω → {∅} ∈ 𝐴)
97 elpri 3550 . . . 4 ({∅} ∈ {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}} → ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
9897, 2eleq2s 2234 . . 3 ({∅} ∈ 𝐴 → ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
9996, 98syl 14 . 2 (𝐴 ∈ ω → ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
10013snid 3556 . . . . . . 7 ∅ ∈ {∅}
101 eleq2 2203 . . . . . . 7 ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} → (∅ ∈ {∅} ↔ ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑}))
102100, 101mpbii 147 . . . . . 6 ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} → ∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑})
103 biidd 171 . . . . . . 7 (𝑥 = ∅ → (𝜑𝜑))
104103elrab 2840 . . . . . 6 (∅ ∈ {𝑥 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {∅} ∧ 𝜑))
105102, 104sylib 121 . . . . 5 ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} → (∅ ∈ {∅} ∧ 𝜑))
106105simprd 113 . . . 4 ({∅} = {𝑥 ∈ {∅} ∣ 𝜑} → 𝜑)
107 eleq2 2203 . . . . . . 7 ({∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (∅ ∈ {∅} ↔ ∅ ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑}))
108100, 107mpbii 147 . . . . . 6 ({∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ∅ ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑})
109 biidd 171 . . . . . . 7 (𝑥 = ∅ → (¬ 𝜑 ↔ ¬ 𝜑))
110109elrab 2840 . . . . . 6 (∅ ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ↔ (∅ ∈ {∅} ∧ ¬ 𝜑))
111108, 110sylib 121 . . . . 5 ({∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (∅ ∈ {∅} ∧ ¬ 𝜑))
112111simprd 113 . . . 4 ({∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ¬ 𝜑)
113106, 112orim12i 748 . . 3 (({∅} = {𝑥 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}) → (𝜑 ∨ ¬ 𝜑))
114 df-dc 820 . . 3 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
115113, 114sylibr 133 . 2 (({∅} = {𝑥 ∈ {∅} ∣ 𝜑} ∨ {∅} = {𝑥 ∈ {∅} ∣ ¬ 𝜑}) → DECID 𝜑)
11689, 99, 1153syl 17 1 (ω = (On ∩ Fin) → DECID 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  DECID wdc 819   = wceq 1331  ∃wex 1468   ∈ wcel 1480   ≠ wne 2308  ∀wral 2416  {crab 2420  Vcvv 2686   ∩ cin 3070  ∅c0 3363  {csn 3527  {cpr 3528   class class class wbr 3929  Tr wtr 4026  Ord word 4284  Oncon0 4285  ωcom 4504  1oc1o 6306  2oc2o 6307   ≈ cen 6632  Fincfn 6634 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502 This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1o 6313  df-2o 6314  df-er 6429  df-en 6635  df-fin 6637 This theorem is referenced by:  exmidonfin  7050
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