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Theorem undifexmid 4194
Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3504 and undifdcss 6922 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 4131 . . . . 5 ∅ ∈ V
21snid 3624 . . . 4 ∅ ∈ {∅}
3 ssrab2 3241 . . . . 5 {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
4 p0ex 4189 . . . . . . 7 {∅} ∈ V
54rabex 4148 . . . . . 6 {𝑧 ∈ {∅} ∣ 𝜑} ∈ V
6 sseq12 3181 . . . . . . 7 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑥𝑦 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
7 simpl 109 . . . . . . . . 9 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → 𝑥 = {𝑧 ∈ {∅} ∣ 𝜑})
8 simpr 110 . . . . . . . . . 10 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → 𝑦 = {∅})
98, 7difeq12d 3255 . . . . . . . . 9 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑦𝑥) = ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
107, 9uneq12d 3291 . . . . . . . 8 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → (𝑥 ∪ (𝑦𝑥)) = ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})))
1110, 8eqeq12d 2192 . . . . . . 7 ((𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} ∧ 𝑦 = {∅}) → ((𝑥 ∪ (𝑦𝑥)) = 𝑦 ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅}))
126, 11bibi12d 235 . . . . . 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 2793 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅})
153, 14mpbi 145 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) = {∅}
162, 15eleqtrri 2253 . . 3 ∅ ∈ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
17 elun 3277 . . 3 (∅ ∈ ({𝑧 ∈ {∅} ∣ 𝜑} ∪ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})) ↔ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑})))
1816, 17mpbi 145 . 2 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}))
19 biidd 172 . . . . . 6 (𝑧 = ∅ → (𝜑𝜑))
2019elrab3 2895 . . . . 5 (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
212, 20ax-mp 5 . . . 4 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
2221biimpi 120 . . 3 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
23 eldifn 3259 . . . 4 (∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
2423, 21sylnib 676 . . 3 (∅ ∈ ({∅} ∖ {𝑧 ∈ {∅} ∣ 𝜑}) → ¬ 𝜑)
2522, 24orim12i 759 . 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 104  wb 105  wo 708   = wceq 1353  wcel 2148  {crab 2459  cdif 3127  cun 3128  wss 3130  c0 3423  {csn 3593
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 4122  ax-nul 4130  ax-pow 4175
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599
This theorem is referenced by: (None)
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