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Theorem exmid1stab 13184
Description: If any proposition is stable, excluded middle follows. We are thinking of 𝑥 as a proposition and 𝑥 = {∅} as "x is true". (Contributed by Jim Kingdon, 28-Nov-2023.)
Hypothesis
Ref Expression
exmid1stab.x ((𝜑𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})
Assertion
Ref Expression
exmid1stab (𝜑EXMID)
Distinct variable group:   𝜑,𝑥

Proof of Theorem exmid1stab
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 0ex 4050 . . . . . . . . 9 ∅ ∈ V
21snid 3551 . . . . . . . 8 ∅ ∈ {∅}
3 nnexmid 835 . . . . . . . . . . 11 ¬ ¬ (𝑦 = {∅} ∨ ¬ 𝑦 = {∅})
4 uneq1 3218 . . . . . . . . . . . . . . 15 (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = ({∅} ∪ ({∅} ∖ 𝑦)))
5 undifabs 3434 . . . . . . . . . . . . . . 15 ({∅} ∪ ({∅} ∖ 𝑦)) = {∅}
64, 5syl6eq 2186 . . . . . . . . . . . . . 14 (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
76a1i 9 . . . . . . . . . . . . 13 (𝑦 ⊆ {∅} → (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
8 df-ne 2307 . . . . . . . . . . . . . . . . 17 (𝑦 ≠ {∅} ↔ ¬ 𝑦 = {∅})
9 pwntru 4117 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ {∅} ∧ 𝑦 ≠ {∅}) → 𝑦 = ∅)
108, 9sylan2br 286 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → 𝑦 = ∅)
1110difeq2d 3189 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → ({∅} ∖ 𝑦) = ({∅} ∖ ∅))
12 dif0 3428 . . . . . . . . . . . . . . . . 17 ({∅} ∖ ∅) = {∅}
1311, 12syl6eq 2186 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → ({∅} ∖ 𝑦) = {∅})
1410, 13uneq12d 3226 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = (∅ ∪ {∅}))
15 uncom 3215 . . . . . . . . . . . . . . . 16 (∅ ∪ {∅}) = ({∅} ∪ ∅)
16 un0 3391 . . . . . . . . . . . . . . . 16 ({∅} ∪ ∅) = {∅}
1715, 16eqtri 2158 . . . . . . . . . . . . . . 15 (∅ ∪ {∅}) = {∅}
1814, 17syl6eq 2186 . . . . . . . . . . . . . 14 ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
1918ex 114 . . . . . . . . . . . . 13 (𝑦 ⊆ {∅} → (¬ 𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
207, 19jaod 706 . . . . . . . . . . . 12 (𝑦 ⊆ {∅} → ((𝑦 = {∅} ∨ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
2120con3d 620 . . . . . . . . . . 11 (𝑦 ⊆ {∅} → (¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} → ¬ (𝑦 = {∅} ∨ ¬ 𝑦 = {∅})))
223, 21mtoi 653 . . . . . . . . . 10 (𝑦 ⊆ {∅} → ¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
2322adantl 275 . . . . . . . . 9 ((𝜑𝑦 ⊆ {∅}) → ¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
24 difss 3197 . . . . . . . . . . . . 13 ({∅} ∖ 𝑦) ⊆ {∅}
25 unss 3245 . . . . . . . . . . . . . 14 ((𝑦 ⊆ {∅} ∧ ({∅} ∖ 𝑦) ⊆ {∅}) ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
2625biimpi 119 . . . . . . . . . . . . 13 ((𝑦 ⊆ {∅} ∧ ({∅} ∖ 𝑦) ⊆ {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
2724, 26mpan2 421 . . . . . . . . . . . 12 (𝑦 ⊆ {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
2827adantl 275 . . . . . . . . . . 11 ((𝜑𝑦 ⊆ {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
29 exmid1stab.x . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})
3029ex 114 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ⊆ {∅} → STAB 𝑥 = {∅}))
3130alrimiv 1846 . . . . . . . . . . . 12 (𝜑 → ∀𝑥(𝑥 ⊆ {∅} → STAB 𝑥 = {∅}))
32 p0ex 4107 . . . . . . . . . . . . . 14 {∅} ∈ V
3332ssex 4060 . . . . . . . . . . . . 13 ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) ∈ V)
34 sseq1 3115 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (𝑥 ⊆ {∅} ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅}))
35 eqeq1 2144 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (𝑥 = {∅} ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
3635stbid 817 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (STAB 𝑥 = {∅} ↔ STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
3734, 36imbi12d 233 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → ((𝑥 ⊆ {∅} → STAB 𝑥 = {∅}) ↔ ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})))
3837spcgv 2768 . . . . . . . . . . . . 13 ((𝑦 ∪ ({∅} ∖ 𝑦)) ∈ V → (∀𝑥(𝑥 ⊆ {∅} → STAB 𝑥 = {∅}) → ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})))
3927, 33, 383syl 17 . . . . . . . . . . . 12 (𝑦 ⊆ {∅} → (∀𝑥(𝑥 ⊆ {∅} → STAB 𝑥 = {∅}) → ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})))
4031, 39mpan9 279 . . . . . . . . . . 11 ((𝜑𝑦 ⊆ {∅}) → ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
4128, 40mpd 13 . . . . . . . . . 10 ((𝜑𝑦 ⊆ {∅}) → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
42 df-stab 816 . . . . . . . . . 10 (STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} ↔ (¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
4341, 42sylib 121 . . . . . . . . 9 ((𝜑𝑦 ⊆ {∅}) → (¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
4423, 43mpd 13 . . . . . . . 8 ((𝜑𝑦 ⊆ {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
452, 44eleqtrrid 2227 . . . . . . 7 ((𝜑𝑦 ⊆ {∅}) → ∅ ∈ (𝑦 ∪ ({∅} ∖ 𝑦)))
46 elun 3212 . . . . . . 7 (∅ ∈ (𝑦 ∪ ({∅} ∖ 𝑦)) ↔ (∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)))
4745, 46sylib 121 . . . . . 6 ((𝜑𝑦 ⊆ {∅}) → (∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)))
48 eldifn 3194 . . . . . . 7 (∅ ∈ ({∅} ∖ 𝑦) → ¬ ∅ ∈ 𝑦)
4948orim2i 750 . . . . . 6 ((∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)) → (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
5047, 49syl 14 . . . . 5 ((𝜑𝑦 ⊆ {∅}) → (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
51 df-dc 820 . . . . 5 (DECID ∅ ∈ 𝑦 ↔ (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
5250, 51sylibr 133 . . . 4 ((𝜑𝑦 ⊆ {∅}) → DECID ∅ ∈ 𝑦)
5352ex 114 . . 3 (𝜑 → (𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
5453alrimiv 1846 . 2 (𝜑 → ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
55 df-exmid 4114 . 2 (EXMID ↔ ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
5654, 55sylibr 133 1 (𝜑EXMID)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 697  STAB wstab 815  DECID wdc 819  wal 1329   = wceq 1331  wcel 1480  wne 2306  Vcvv 2681  cdif 3063  cun 3064  wss 3066  c0 3358  {csn 3522  EXMIDwem 4113
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-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-exmid 4114
This theorem is referenced by:  subctctexmid  13185
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