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Theorem onsucsssucexmid 4169
Description: The converse of onsucsssucr 4157 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 3003 . . . . . 6 {z {∅} ∣ φ} ⊆ {∅}
2 ordtriexmidlem 4165 . . . . . . 7 {z {∅} ∣ φ} On
3 sseq1 2944 . . . . . . . . 9 (x = {z {∅} ∣ φ} → (x ⊆ {∅} ↔ {z {∅} ∣ φ} ⊆ {∅}))
4 suceq 4062 . . . . . . . . . 10 (x = {z {∅} ∣ φ} → suc x = suc {z {∅} ∣ φ})
54sseq1d 2950 . . . . . . . . 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 4071 . . . . . . . . . 10 suc ∅ = {∅}
8 0elon 4052 . . . . . . . . . . 11 On
98onsuci 4164 . . . . . . . . . 10 suc ∅ On
107, 9eqeltrri 2093 . . . . . . . . 9 {∅} On
11 p0ex 3891 . . . . . . . . . 10 {∅} V
12 eleq1 2082 . . . . . . . . . . . 12 (y = {∅} → (y On ↔ {∅} On))
1312anbi2d 440 . . . . . . . . . . 11 (y = {∅} → ((x On y On) ↔ (x On {∅} On)))
14 sseq2 2945 . . . . . . . . . . . 12 (y = {∅} → (xyx ⊆ {∅}))
15 suceq 4062 . . . . . . . . . . . . 13 (y = {∅} → suc y = suc {∅})
1615sseq2d 2951 . . . . . . . . . . . 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 2384 . . . . . . . . . 10 ((x On y On) → (xy → suc x ⊆ suc y))
2111, 18, 20vtocl 2583 . . . . . . . . 9 ((x On {∅} On) → (x ⊆ {∅} → suc x ⊆ suc {∅}))
2210, 21mpan2 403 . . . . . . . 8 (x On → (x ⊆ {∅} → suc x ⊆ suc {∅}))
236, 22vtoclga 2594 . . . . . . 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 4164 . . . . . . 7 suc {∅} On
2726onordi 4086 . . . . . 6 Ord suc {∅}
28 ordelsuc 4154 . . . . . 6 (({z {∅} ∣ φ} On Ord suc {∅}) → ({z {∅} ∣ φ} suc {∅} ↔ suc {z {∅} ∣ φ} ⊆ suc {∅}))
292, 27, 28mp2an 404 . . . . 5 ({z {∅} ∣ φ} suc {∅} ↔ suc {z {∅} ∣ φ} ⊆ suc {∅})
3025, 29mpbir 134 . . . 4 {z {∅} ∣ φ} suc {∅}
31 elsucg 4064 . . . . 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 3351 . . . . 5 ({z {∅} ∣ φ} {∅} → {z {∅} ∣ φ} = ∅)
35 ordtriexmidlem2 4166 . . . . 5 ({z {∅} ∣ φ} = ∅ → ¬ φ)
3634, 35syl 14 . . . 4 ({z {∅} ∣ φ} {∅} → ¬ φ)
37 0ex 3836 . . . . 5 V
38 biidd 161 . . . . 5 (z = ∅ → (φφ))
3937, 38rabsnt 3396 . . . 4 ({z {∅} ∣ φ} = {∅} → φ)
4036, 39orim12i 664 . . 3 (({z {∅} ∣ φ} {∅} {z {∅} ∣ φ} = {∅}) → (¬ φ φ))
4133, 40ax-mp 7 . 2 φ φ)
42 orcom 634 . 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 616   = wceq 1373   wcel 1375  wral 2282  {crab 2286  wss 2895  c0 3202  {csn 3327  Ord word 4023  Oncon0 4024  suc csuc 4026
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 532  ax-in2 533  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-nul 3835  ax-pow 3879  ax-pr 3896  ax-un 4093
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-rab 2291  df-v 2535  df-dif 2898  df-un 2900  df-in 2902  df-ss 2909  df-nul 3203  df-pw 3313  df-sn 3333  df-pr 3334  df-uni 3533  df-tr 3807  df-iord 4027  df-on 4028  df-suc 4031
This theorem is referenced by: (None)
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