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Theorem undifexmid 4126
 Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3449 and undifdcss 6824 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.)
Hypothesis
Ref Expression
undifexmid.1 (𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦)
Assertion
Ref Expression
undifexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem undifexmid
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 0ex 4064 . . . . 5 ∅ ∈ V
21snid 3564 . . . 4 ∅ ∈ {∅}
3 ssrab2 3188 . . . . 5 {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
4 p0ex 4121 . . . . . . 7 {∅} ∈ V
54rabex 4081 . . . . . 6 {𝑧 ∈ {∅} ∣ 𝜑} ∈ V
6 sseq12 3128 . . . . . . 7 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑥𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
7 simpl 108 . . . . . . . . 9 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → 𝑥 = {𝑧 ∈ {∅} ∣ 𝜑})
8 simpr 109 . . . . . . . . . 10 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → 𝑦 = {∅})
98, 7difeq12d 3201 . . . . . . . . 9 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑦𝑥) = ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
107, 9uneq12d 3237 . . . . . . . 8 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑥 ∪ (𝑦𝑥)) = ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})))
1110, 8eqeq12d 2155 . . . . . . 7 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → ((𝑥 ∪ (𝑦𝑥)) = 𝑦 ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅}))
126, 11bibi12d 234 . . . . . 6 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → ((𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅})))
13 undifexmid.1 . . . . . 6 (𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦)
145, 4, 12, 13vtocl2 2745 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅})
153, 14mpbi 144 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅}
162, 15eleqtrri 2216 . . 3 ∅ ∈ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
17 elun 3223 . . 3 (∅ ∈ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) ↔ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})))
1816, 17mpbi 144 . 2 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
19 biidd 171 . . . . . 6 (𝑧 = ∅ → (𝜑𝜑))
2019elrab3 2846 . . . . 5 (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
212, 20ax-mp 5 . . . 4 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
2221biimpi 119 . . 3 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
23 eldifn 3205 . . . 4 (∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
2423, 21sylnib 666 . . 3 (∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ 𝜑)
2522, 24orim12i 749 . 2 ((∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) → (𝜑 ∨ ¬ 𝜑))
2618, 25ax-mp 5 1 (𝜑 ∨ ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1332   ∈ wcel 1481  {crab 2421   ∖ cdif 3074   ∪ cun 3075   ⊆ wss 3077  ∅c0 3369  {csn 3533 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-nul 4063  ax-pow 4107 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2692  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-pw 3518  df-sn 3539 This theorem is referenced by: (None)
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