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Theorem exmidsssn 4203
Description: Excluded middle is equivalent to the biconditionalized version of sssnr 3754 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.)
Assertion
Ref Expression
exmidsssn (EXMID ↔ ∀𝑥𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidsssn
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 0ss 3462 . . . . . . 7 ∅ ⊆ {𝑦}
2 sseq1 3179 . . . . . . 7 (𝑥 = ∅ → (𝑥 ⊆ {𝑦} ↔ ∅ ⊆ {𝑦}))
31, 2mpbiri 168 . . . . . 6 (𝑥 = ∅ → 𝑥 ⊆ {𝑦})
43adantl 277 . . . . 5 ((EXMID𝑥 = ∅) → 𝑥 ⊆ {𝑦})
5 simpr 110 . . . . . 6 ((EXMID𝑥 = ∅) → 𝑥 = ∅)
65orcd 733 . . . . 5 ((EXMID𝑥 = ∅) → (𝑥 = ∅ ∨ 𝑥 = {𝑦}))
74, 62thd 175 . . . 4 ((EXMID𝑥 = ∅) → (𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
8 sssnm 3755 . . . . . 6 (∃𝑧 𝑧𝑥 → (𝑥 ⊆ {𝑦} ↔ 𝑥 = {𝑦}))
9 neq0r 3438 . . . . . . 7 (∃𝑧 𝑧𝑥 → ¬ 𝑥 = ∅)
10 biorf 744 . . . . . . 7 𝑥 = ∅ → (𝑥 = {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
119, 10syl 14 . . . . . 6 (∃𝑧 𝑧𝑥 → (𝑥 = {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
128, 11bitrd 188 . . . . 5 (∃𝑧 𝑧𝑥 → (𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
1312adantl 277 . . . 4 ((EXMID ∧ ∃𝑧 𝑧𝑥) → (𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
14 exmidn0m 4202 . . . . . 6 (EXMID ↔ ∀𝑥(𝑥 = ∅ ∨ ∃𝑧 𝑧𝑥))
1514biimpi 120 . . . . 5 (EXMID → ∀𝑥(𝑥 = ∅ ∨ ∃𝑧 𝑧𝑥))
161519.21bi 1558 . . . 4 (EXMID → (𝑥 = ∅ ∨ ∃𝑧 𝑧𝑥))
177, 13, 16mpjaodan 798 . . 3 (EXMID → (𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
1817alrimivv 1875 . 2 (EXMID → ∀𝑥𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
19 0ex 4131 . . . . . 6 ∅ ∈ V
20 sneq 3604 . . . . . . . 8 (𝑦 = ∅ → {𝑦} = {∅})
2120sseq2d 3186 . . . . . . 7 (𝑦 = ∅ → (𝑥 ⊆ {𝑦} ↔ 𝑥 ⊆ {∅}))
2220eqeq2d 2189 . . . . . . . 8 (𝑦 = ∅ → (𝑥 = {𝑦} ↔ 𝑥 = {∅}))
2322orbi2d 790 . . . . . . 7 (𝑦 = ∅ → ((𝑥 = ∅ ∨ 𝑥 = {𝑦}) ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
2421, 23bibi12d 235 . . . . . 6 (𝑦 = ∅ → ((𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})) ↔ (𝑥 ⊆ {∅} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))))
2519, 24spcv 2832 . . . . 5 (∀𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})) → (𝑥 ⊆ {∅} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
2625biimpd 144 . . . 4 (∀𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})) → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2726alimi 1455 . . 3 (∀𝑥𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})) → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
28 exmid01 4199 . . 3 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2927, 28sylibr 134 . 2 (∀𝑥𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})) → EXMID)
3018, 29impbii 126 1 (EXMID ↔ ∀𝑥𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  wal 1351   = wceq 1353  wex 1492  wss 3130  c0 3423  {csn 3593  EXMIDwem 4195
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-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-rab 2464  df-v 2740  df-dif 3132  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-exmid 4196
This theorem is referenced by:  exmidsssnc  4204
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