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Theorem exmid1stab 13559
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 4091 . . . . . . . . 9 ∅ ∈ V
21snid 3591 . . . . . . . 8 ∅ ∈ {∅}
3 nnexmid 836 . . . . . . . . . . 11 ¬ ¬ (𝑦 = {∅} ∨ ¬ 𝑦 = {∅})
4 uneq1 3254 . . . . . . . . . . . . . . 15 (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = ({∅} ∪ ({∅} ∖ 𝑦)))
5 undifabs 3470 . . . . . . . . . . . . . . 15 ({∅} ∪ ({∅} ∖ 𝑦)) = {∅}
64, 5eqtrdi 2206 . . . . . . . . . . . . . 14 (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
76a1i 9 . . . . . . . . . . . . 13 (𝑦 ⊆ {∅} → (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
8 df-ne 2328 . . . . . . . . . . . . . . . . 17 (𝑦 ≠ {∅} ↔ ¬ 𝑦 = {∅})
9 pwntru 4160 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ {∅} ∧ 𝑦 ≠ {∅}) → 𝑦 = ∅)
108, 9sylan2br 286 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → 𝑦 = ∅)
1110difeq2d 3225 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → ({∅} ∖ 𝑦) = ({∅} ∖ ∅))
12 dif0 3464 . . . . . . . . . . . . . . . . 17 ({∅} ∖ ∅) = {∅}
1311, 12eqtrdi 2206 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → ({∅} ∖ 𝑦) = {∅})
1410, 13uneq12d 3262 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = (∅ ∪ {∅}))
15 uncom 3251 . . . . . . . . . . . . . . . 16 (∅ ∪ {∅}) = ({∅} ∪ ∅)
16 un0 3427 . . . . . . . . . . . . . . . 16 ({∅} ∪ ∅) = {∅}
1715, 16eqtri 2178 . . . . . . . . . . . . . . 15 (∅ ∪ {∅}) = {∅}
1814, 17eqtrdi 2206 . . . . . . . . . . . . . 14 ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
1918ex 114 . . . . . . . . . . . . 13 (𝑦 ⊆ {∅} → (¬ 𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
207, 19jaod 707 . . . . . . . . . . . 12 (𝑦 ⊆ {∅} → ((𝑦 = {∅} ∨ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
2120con3d 621 . . . . . . . . . . 11 (𝑦 ⊆ {∅} → (¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} → ¬ (𝑦 = {∅} ∨ ¬ 𝑦 = {∅})))
223, 21mtoi 654 . . . . . . . . . 10 (𝑦 ⊆ {∅} → ¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
2322adantl 275 . . . . . . . . 9 ((𝜑𝑦 ⊆ {∅}) → ¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
24 difss 3233 . . . . . . . . . . . . 13 ({∅} ∖ 𝑦) ⊆ {∅}
25 unss 3281 . . . . . . . . . . . . . 14 ((𝑦 ⊆ {∅} ∧ ({∅} ∖ 𝑦) ⊆ {∅}) ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
2625biimpi 119 . . . . . . . . . . . . 13 ((𝑦 ⊆ {∅} ∧ ({∅} ∖ 𝑦) ⊆ {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
2724, 26mpan2 422 . . . . . . . . . . . 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 1854 . . . . . . . . . . . 12 (𝜑 → ∀𝑥(𝑥 ⊆ {∅} → STAB 𝑥 = {∅}))
32 p0ex 4149 . . . . . . . . . . . . . 14 {∅} ∈ V
3332ssex 4101 . . . . . . . . . . . . 13 ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) ∈ V)
34 sseq1 3151 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (𝑥 ⊆ {∅} ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅}))
35 eqeq1 2164 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (𝑥 = {∅} ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
3635stbid 818 . . . . . . . . . . . . . . 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 2799 . . . . . . . . . . . . 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 817 . . . . . . . . . 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 2247 . . . . . . 7 ((𝜑𝑦 ⊆ {∅}) → ∅ ∈ (𝑦 ∪ ({∅} ∖ 𝑦)))
46 elun 3248 . . . . . . 7 (∅ ∈ (𝑦 ∪ ({∅} ∖ 𝑦)) ↔ (∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)))
4745, 46sylib 121 . . . . . 6 ((𝜑𝑦 ⊆ {∅}) → (∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)))
48 eldifn 3230 . . . . . . 7 (∅ ∈ ({∅} ∖ 𝑦) → ¬ ∅ ∈ 𝑦)
4948orim2i 751 . . . . . 6 ((∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)) → (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
5047, 49syl 14 . . . . 5 ((𝜑𝑦 ⊆ {∅}) → (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
51 df-dc 821 . . . . 5 (DECID ∅ ∈ 𝑦 ↔ (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
5250, 51sylibr 133 . . . 4 ((𝜑𝑦 ⊆ {∅}) → DECID ∅ ∈ 𝑦)
5352ex 114 . . 3 (𝜑 → (𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
5453alrimiv 1854 . 2 (𝜑 → ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
55 df-exmid 4156 . 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 698  STAB wstab 816  DECID wdc 820  wal 1333   = wceq 1335  wcel 2128  wne 2327  Vcvv 2712  cdif 3099  cun 3100  wss 3102  c0 3394  {csn 3560  EXMIDwem 4155
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-exmid 4156
This theorem is referenced by:  subctctexmid  13560
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