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Theorem ordtri2orexmid 4193
Description: Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.)
Hypothesis
Ref Expression
ordtri2orexmid.1 x On y On (x y yx)
Assertion
Ref Expression
ordtri2orexmid (φ ¬ φ)
Distinct variable group:   φ,x,y

Proof of Theorem ordtri2orexmid
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 ordtri2orexmid.1 . . . 4 x On y On (x y yx)
2 ordtriexmidlem 4190 . . . . 5 {z {∅} ∣ φ} On
3 suc0 4095 . . . . . 6 suc ∅ = {∅}
4 0elon 4076 . . . . . . 7 On
54onsuci 4189 . . . . . 6 suc ∅ On
63, 5eqeltrri 2094 . . . . 5 {∅} On
7 eleq1 2083 . . . . . . 7 (x = {z {∅} ∣ φ} → (x y ↔ {z {∅} ∣ φ} y))
8 sseq2 2943 . . . . . . 7 (x = {z {∅} ∣ φ} → (yxy ⊆ {z {∅} ∣ φ}))
97, 8orbi12d 694 . . . . . 6 (x = {z {∅} ∣ φ} → ((x y yx) ↔ ({z {∅} ∣ φ} y y ⊆ {z {∅} ∣ φ})))
10 eleq2 2084 . . . . . . 7 (y = {∅} → ({z {∅} ∣ φ} y ↔ {z {∅} ∣ φ} {∅}))
11 sseq1 2942 . . . . . . 7 (y = {∅} → (y ⊆ {z {∅} ∣ φ} ↔ {∅} ⊆ {z {∅} ∣ φ}))
1210, 11orbi12d 694 . . . . . 6 (y = {∅} → (({z {∅} ∣ φ} y y ⊆ {z {∅} ∣ φ}) ↔ ({z {∅} ∣ φ} {∅} {∅} ⊆ {z {∅} ∣ φ})))
139, 12rspc2va 2639 . . . . 5 ((({z {∅} ∣ φ} On {∅} On) x On y On (x y yx)) → ({z {∅} ∣ φ} {∅} {∅} ⊆ {z {∅} ∣ φ}))
142, 6, 13mpanl12 414 . . . 4 (x On y On (x y yx) → ({z {∅} ∣ φ} {∅} {∅} ⊆ {z {∅} ∣ φ}))
151, 14ax-mp 7 . . 3 ({z {∅} ∣ φ} {∅} {∅} ⊆ {z {∅} ∣ φ})
16 elsni 3373 . . . . 5 ({z {∅} ∣ φ} {∅} → {z {∅} ∣ φ} = ∅)
17 ordtriexmidlem2 4191 . . . . 5 ({z {∅} ∣ φ} = ∅ → ¬ φ)
1816, 17syl 14 . . . 4 ({z {∅} ∣ φ} {∅} → ¬ φ)
19 snssg 3473 . . . . . 6 (∅ On → (∅ {z {∅} ∣ φ} ↔ {∅} ⊆ {z {∅} ∣ φ}))
204, 19ax-mp 7 . . . . 5 (∅ {z {∅} ∣ φ} ↔ {∅} ⊆ {z {∅} ∣ φ})
21 0ex 3857 . . . . . . . 8 V
2221snid 3376 . . . . . . 7 {∅}
23 biidd 161 . . . . . . . 8 (z = ∅ → (φφ))
2423elrab3 2675 . . . . . . 7 (∅ {∅} → (∅ {z {∅} ∣ φ} ↔ φ))
2522, 24ax-mp 7 . . . . . 6 (∅ {z {∅} ∣ φ} ↔ φ)
2625biimpi 113 . . . . 5 (∅ {z {∅} ∣ φ} → φ)
2720, 26sylbir 125 . . . 4 ({∅} ⊆ {z {∅} ∣ φ} → φ)
2818, 27orim12i 663 . . 3 (({z {∅} ∣ φ} {∅} {∅} ⊆ {z {∅} ∣ φ}) → (¬ φ φ))
2915, 28ax-mp 7 . 2 φ φ)
30 orcom 634 . 2 ((¬ φ φ) ↔ (φ ¬ φ))
3129, 30mpbi 133 1 (φ ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98   wo 616   = wceq 1228   wcel 1375  wral 2283  {crab 2287  wss 2893  c0 3200  {csn 3349  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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