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Theorem onsucsssucexmid 4211
Description: The converse of onsucsssucr 4199 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
Hypothesis
Ref Expression
onsucsssucexmid.1 x On y On (xy → suc x ⊆ suc y)
Assertion
Ref Expression
onsucsssucexmid (φ ¬ φ)
Distinct variable groups:   φ,x   x,y
Allowed substitution hint:   φ(y)

Proof of Theorem onsucsssucexmid
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 ssrab2 3019 . . . . . 6 {z {∅} ∣ φ} ⊆ {∅}
2 ordtriexmidlem 4207 . . . . . . 7 {z {∅} ∣ φ} On
3 sseq1 2960 . . . . . . . . 9 (x = {z {∅} ∣ φ} → (x ⊆ {∅} ↔ {z {∅} ∣ φ} ⊆ {∅}))
4 suceq 4104 . . . . . . . . . 10 (x = {z {∅} ∣ φ} → suc x = suc {z {∅} ∣ φ})
54sseq1d 2966 . . . . . . . . 9 (x = {z {∅} ∣ φ} → (suc x ⊆ suc {∅} ↔ suc {z {∅} ∣ φ} ⊆ suc {∅}))
63, 5imbi12d 223 . . . . . . . 8 (x = {z {∅} ∣ φ} → ((x ⊆ {∅} → suc x ⊆ suc {∅}) ↔ ({z {∅} ∣ φ} ⊆ {∅} → suc {z {∅} ∣ φ} ⊆ suc {∅})))
7 suc0 4113 . . . . . . . . . 10 suc ∅ = {∅}
8 0elon 4094 . . . . . . . . . . 11 On
98onsuci 4206 . . . . . . . . . 10 suc ∅ On
107, 9eqeltrri 2108 . . . . . . . . 9 {∅} On
11 p0ex 3929 . . . . . . . . . 10 {∅} V
12 eleq1 2097 . . . . . . . . . . . 12 (y = {∅} → (y On ↔ {∅} On))
1312anbi2d 437 . . . . . . . . . . 11 (y = {∅} → ((x On y On) ↔ (x On {∅} On)))
14 sseq2 2961 . . . . . . . . . . . 12 (y = {∅} → (xyx ⊆ {∅}))
15 suceq 4104 . . . . . . . . . . . . 13 (y = {∅} → suc y = suc {∅})
1615sseq2d 2967 . . . . . . . . . . . 12 (y = {∅} → (suc x ⊆ suc y ↔ suc x ⊆ suc {∅}))
1714, 16imbi12d 223 . . . . . . . . . . 11 (y = {∅} → ((xy → suc x ⊆ suc y) ↔ (x ⊆ {∅} → suc x ⊆ suc {∅})))
1813, 17imbi12d 223 . . . . . . . . . 10 (y = {∅} → (((x On y On) → (xy → suc x ⊆ suc y)) ↔ ((x On {∅} On) → (x ⊆ {∅} → suc x ⊆ suc {∅}))))
19 onsucsssucexmid.1 . . . . . . . . . . 11 x On y On (xy → suc x ⊆ suc y)
2019rspec2 2402 . . . . . . . . . 10 ((x On y On) → (xy → suc x ⊆ suc y))
2111, 18, 20vtocl 2602 . . . . . . . . 9 ((x On {∅} On) → (x ⊆ {∅} → suc x ⊆ suc {∅}))
2210, 21mpan2 401 . . . . . . . 8 (x On → (x ⊆ {∅} → suc x ⊆ suc {∅}))
236, 22vtoclga 2613 . . . . . . 7 ({z {∅} ∣ φ} On → ({z {∅} ∣ φ} ⊆ {∅} → suc {z {∅} ∣ φ} ⊆ suc {∅}))
242, 23ax-mp 7 . . . . . 6 ({z {∅} ∣ φ} ⊆ {∅} → suc {z {∅} ∣ φ} ⊆ suc {∅})
251, 24ax-mp 7 . . . . 5 suc {z {∅} ∣ φ} ⊆ suc {∅}
2610onsuci 4206 . . . . . . 7 suc {∅} On
2726onordi 4128 . . . . . 6 Ord suc {∅}
28 ordelsuc 4196 . . . . . 6 (({z {∅} ∣ φ} On Ord suc {∅}) → ({z {∅} ∣ φ} suc {∅} ↔ suc {z {∅} ∣ φ} ⊆ suc {∅}))
292, 27, 28mp2an 402 . . . . 5 ({z {∅} ∣ φ} suc {∅} ↔ suc {z {∅} ∣ φ} ⊆ suc {∅})
3025, 29mpbir 134 . . . 4 {z {∅} ∣ φ} suc {∅}
31 elsucg 4106 . . . . 5 ({z {∅} ∣ φ} On → ({z {∅} ∣ φ} suc {∅} ↔ ({z {∅} ∣ φ} {∅} {z {∅} ∣ φ} = {∅})))
322, 31ax-mp 7 . . . 4 ({z {∅} ∣ φ} suc {∅} ↔ ({z {∅} ∣ φ} {∅} {z {∅} ∣ φ} = {∅}))
3330, 32mpbi 133 . . 3 ({z {∅} ∣ φ} {∅} {z {∅} ∣ φ} = {∅})
34 elsni 3390 . . . . 5 ({z {∅} ∣ φ} {∅} → {z {∅} ∣ φ} = ∅)
35 ordtriexmidlem2 4208 . . . . 5 ({z {∅} ∣ φ} = ∅ → ¬ φ)
3634, 35syl 14 . . . 4 ({z {∅} ∣ φ} {∅} → ¬ φ)
37 0ex 3874 . . . . 5 V
38 biidd 161 . . . . 5 (z = ∅ → (φφ))
3937, 38rabsnt 3435 . . . 4 ({z {∅} ∣ φ} = {∅} → φ)
4036, 39orim12i 675 . . 3 (({z {∅} ∣ φ} {∅} {z {∅} ∣ φ} = {∅}) → (¬ φ φ))
4133, 40ax-mp 7 . 2 φ φ)
42 orcom 646 . 2 ((¬ φ φ) ↔ (φ ¬ φ))
4341, 42mpbi 133 1 (φ ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628   = wceq 1242   wcel 1390  wral 2300  {crab 2304  wss 2911  c0 3218  {csn 3366  Ord word 4064  Oncon0 4065  suc csuc 4067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3352  df-sn 3372  df-pr 3373  df-uni 3571  df-tr 3845  df-iord 4068  df-on 4070  df-suc 4073
This theorem is referenced by: (None)
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