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Theorem onsucsssucexmid 4194
 Description: The converse of onsucsssucr 4182 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 3001 . . . . . 6 {z {∅} ∣ φ} ⊆ {∅}
2 ordtriexmidlem 4190 . . . . . . 7 {z {∅} ∣ φ} On
3 sseq1 2942 . . . . . . . . 9 (x = {z {∅} ∣ φ} → (x ⊆ {∅} ↔ {z {∅} ∣ φ} ⊆ {∅}))
4 suceq 4086 . . . . . . . . . 10 (x = {z {∅} ∣ φ} → suc x = suc {z {∅} ∣ φ})
54sseq1d 2948 . . . . . . . . 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 4095 . . . . . . . . . 10 suc ∅ = {∅}
8 0elon 4076 . . . . . . . . . . 11 On
98onsuci 4189 . . . . . . . . . 10 suc ∅ On
107, 9eqeltrri 2094 . . . . . . . . 9 {∅} On
11 p0ex 3912 . . . . . . . . . 10 {∅} V
12 eleq1 2083 . . . . . . . . . . . 12 (y = {∅} → (y On ↔ {∅} On))
1312anbi2d 440 . . . . . . . . . . 11 (y = {∅} → ((x On y On) ↔ (x On {∅} On)))
14 sseq2 2943 . . . . . . . . . . . 12 (y = {∅} → (xyx ⊆ {∅}))
15 suceq 4086 . . . . . . . . . . . . 13 (y = {∅} → suc y = suc {∅})
1615sseq2d 2949 . . . . . . . . . . . 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 2385 . . . . . . . . . 10 ((x On y On) → (xy → suc x ⊆ suc y))
2111, 18, 20vtocl 2584 . . . . . . . . 9 ((x On {∅} On) → (x ⊆ {∅} → suc x ⊆ suc {∅}))
2210, 21mpan2 403 . . . . . . . 8 (x On → (x ⊆ {∅} → suc x ⊆ suc {∅}))
236, 22vtoclga 2595 . . . . . . 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 4189 . . . . . . 7 suc {∅} On
2726onordi 4111 . . . . . 6 Ord suc {∅}
28 ordelsuc 4179 . . . . . 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 4088 . . . . 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 3373 . . . . 5 ({z {∅} ∣ φ} {∅} → {z {∅} ∣ φ} = ∅)
35 ordtriexmidlem2 4191 . . . . 5 ({z {∅} ∣ φ} = ∅ → ¬ φ)
3634, 35syl 14 . . . 4 ({z {∅} ∣ φ} {∅} → ¬ φ)
37 0ex 3857 . . . . 5 V
38 biidd 161 . . . . 5 (z = ∅ → (φφ))
3937, 38rabsnt 3418 . . . 4 ({z {∅} ∣ φ} = {∅} → φ)
4036, 39orim12i 663 . . 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 1228   ∈ wcel 1375  ∀wral 2283  {crab 2287   ⊆ wss 2893  ∅c0 3200  {csn 3349  Ord word 4046  Oncon0 4047  suc csuc 4049 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  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 2005  ax-sep 3848  ax-nul 3856  ax-pow 3900  ax-pr 3917  ax-un 4118 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-rab 2292  df-v 2536  df-dif 2896  df-un 2898  df-in 2900  df-ss 2907  df-nul 3201  df-pw 3335  df-sn 3355  df-pr 3356  df-uni 3554  df-tr 3828  df-iord 4050  df-on 4052  df-suc 4055 This theorem is referenced by: (None)
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