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Theorem exmid1stab 4205
Description: If every 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 4127 . . . . . . . . 9 ∅ ∈ V
21snid 3622 . . . . . . . 8 ∅ ∈ {∅}
3 nnexmid 850 . . . . . . . . . . 11 ¬ ¬ (𝑦 = {∅} ∨ ¬ 𝑦 = {∅})
4 uneq1 3282 . . . . . . . . . . . . . . 15 (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = ({∅} ∪ ({∅} ∖ 𝑦)))
5 undifabs 3499 . . . . . . . . . . . . . . 15 ({∅} ∪ ({∅} ∖ 𝑦)) = {∅}
64, 5eqtrdi 2226 . . . . . . . . . . . . . 14 (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
76a1i 9 . . . . . . . . . . . . 13 (𝑦 ⊆ {∅} → (𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
8 df-ne 2348 . . . . . . . . . . . . . . . . 17 (𝑦 ≠ {∅} ↔ ¬ 𝑦 = {∅})
9 pwntru 4196 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ {∅} ∧ 𝑦 ≠ {∅}) → 𝑦 = ∅)
108, 9sylan2br 288 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → 𝑦 = ∅)
1110difeq2d 3253 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → ({∅} ∖ 𝑦) = ({∅} ∖ ∅))
12 dif0 3493 . . . . . . . . . . . . . . . . 17 ({∅} ∖ ∅) = {∅}
1311, 12eqtrdi 2226 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → ({∅} ∖ 𝑦) = {∅})
1410, 13uneq12d 3290 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = (∅ ∪ {∅}))
15 uncom 3279 . . . . . . . . . . . . . . . 16 (∅ ∪ {∅}) = ({∅} ∪ ∅)
16 un0 3456 . . . . . . . . . . . . . . . 16 ({∅} ∪ ∅) = {∅}
1715, 16eqtri 2198 . . . . . . . . . . . . . . 15 (∅ ∪ {∅}) = {∅}
1814, 17eqtrdi 2226 . . . . . . . . . . . . . 14 ((𝑦 ⊆ {∅} ∧ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
1918ex 115 . . . . . . . . . . . . 13 (𝑦 ⊆ {∅} → (¬ 𝑦 = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
207, 19jaod 717 . . . . . . . . . . . 12 (𝑦 ⊆ {∅} → ((𝑦 = {∅} ∨ ¬ 𝑦 = {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
2120con3d 631 . . . . . . . . . . 11 (𝑦 ⊆ {∅} → (¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} → ¬ (𝑦 = {∅} ∨ ¬ 𝑦 = {∅})))
223, 21mtoi 664 . . . . . . . . . 10 (𝑦 ⊆ {∅} → ¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
2322adantl 277 . . . . . . . . 9 ((𝜑𝑦 ⊆ {∅}) → ¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
24 difss 3261 . . . . . . . . . . . . 13 ({∅} ∖ 𝑦) ⊆ {∅}
25 unss 3309 . . . . . . . . . . . . . 14 ((𝑦 ⊆ {∅} ∧ ({∅} ∖ 𝑦) ⊆ {∅}) ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
2625biimpi 120 . . . . . . . . . . . . 13 ((𝑦 ⊆ {∅} ∧ ({∅} ∖ 𝑦) ⊆ {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
2724, 26mpan2 425 . . . . . . . . . . . 12 (𝑦 ⊆ {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
2827adantl 277 . . . . . . . . . . 11 ((𝜑𝑦 ⊆ {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅})
29 exmid1stab.x . . . . . . . . . . . . . 14 ((𝜑𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})
3029ex 115 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ⊆ {∅} → STAB 𝑥 = {∅}))
3130alrimiv 1874 . . . . . . . . . . . 12 (𝜑 → ∀𝑥(𝑥 ⊆ {∅} → STAB 𝑥 = {∅}))
32 p0ex 4185 . . . . . . . . . . . . . 14 {∅} ∈ V
3332ssex 4137 . . . . . . . . . . . . 13 ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) ∈ V)
34 sseq1 3178 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (𝑥 ⊆ {∅} ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅}))
35 eqeq1 2184 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (𝑥 = {∅} ↔ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
3635stbid 832 . . . . . . . . . . . . . . 15 (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → (STAB 𝑥 = {∅} ↔ STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
3734, 36imbi12d 234 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 ∪ ({∅} ∖ 𝑦)) → ((𝑥 ⊆ {∅} → STAB 𝑥 = {∅}) ↔ ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})))
3837spcgv 2824 . . . . . . . . . . . . 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 281 . . . . . . . . . . 11 ((𝜑𝑦 ⊆ {∅}) → ((𝑦 ∪ ({∅} ∖ 𝑦)) ⊆ {∅} → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
4128, 40mpd 13 . . . . . . . . . 10 ((𝜑𝑦 ⊆ {∅}) → STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
42 df-stab 831 . . . . . . . . . 10 (STAB (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} ↔ (¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
4341, 42sylib 122 . . . . . . . . 9 ((𝜑𝑦 ⊆ {∅}) → (¬ ¬ (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅} → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅}))
4423, 43mpd 13 . . . . . . . 8 ((𝜑𝑦 ⊆ {∅}) → (𝑦 ∪ ({∅} ∖ 𝑦)) = {∅})
452, 44eleqtrrid 2267 . . . . . . 7 ((𝜑𝑦 ⊆ {∅}) → ∅ ∈ (𝑦 ∪ ({∅} ∖ 𝑦)))
46 elun 3276 . . . . . . 7 (∅ ∈ (𝑦 ∪ ({∅} ∖ 𝑦)) ↔ (∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)))
4745, 46sylib 122 . . . . . 6 ((𝜑𝑦 ⊆ {∅}) → (∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)))
48 eldifn 3258 . . . . . . 7 (∅ ∈ ({∅} ∖ 𝑦) → ¬ ∅ ∈ 𝑦)
4948orim2i 761 . . . . . 6 ((∅ ∈ 𝑦 ∨ ∅ ∈ ({∅} ∖ 𝑦)) → (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
5047, 49syl 14 . . . . 5 ((𝜑𝑦 ⊆ {∅}) → (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
51 df-dc 835 . . . . 5 (DECID ∅ ∈ 𝑦 ↔ (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
5250, 51sylibr 134 . . . 4 ((𝜑𝑦 ⊆ {∅}) → DECID ∅ ∈ 𝑦)
5352ex 115 . . 3 (𝜑 → (𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
5453alrimiv 1874 . 2 (𝜑 → ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
55 df-exmid 4192 . 2 (EXMID ↔ ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
5654, 55sylibr 134 1 (𝜑EXMID)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708  STAB wstab 830  DECID wdc 834  wal 1351   = wceq 1353  wcel 2148  wne 2347  Vcvv 2737  cdif 3126  cun 3127  wss 3129  c0 3422  {csn 3591  EXMIDwem 4191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-exmid 4192
This theorem is referenced by:  2omotap  7249  subctctexmid  14406
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