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Theorem ss1oel2o 12161
 Description: Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4038 which more directly illustrates the contrast with el2oss1o 12160. (Contributed by Jim Kingdon, 8-Aug-2022.)
Assertion
Ref Expression
ss1oel2o (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o𝑥 ∈ 2o))

Proof of Theorem ss1oel2o
StepHypRef Expression
1 exmid01 4038 . 2 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2 df1o2 6208 . . . . 5 1o = {∅}
32sseq2i 3052 . . . 4 (𝑥 ⊆ 1o𝑥 ⊆ {∅})
4 df2o2 6210 . . . . . 6 2o = {∅, {∅}}
54eleq2i 2155 . . . . 5 (𝑥 ∈ 2o𝑥 ∈ {∅, {∅}})
6 vex 2623 . . . . . 6 𝑥 ∈ V
76elpr 3471 . . . . 5 (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
85, 7bitri 183 . . . 4 (𝑥 ∈ 2o ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
93, 8imbi12i 238 . . 3 ((𝑥 ⊆ 1o𝑥 ∈ 2o) ↔ (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
109albii 1405 . 2 (∀𝑥(𝑥 ⊆ 1o𝑥 ∈ 2o) ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
111, 10bitr4i 186 1 (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o𝑥 ∈ 2o))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 665  ∀wal 1288   = wceq 1290   ∈ wcel 1439   ⊆ wss 3000  ∅c0 3287  {csn 3450  {cpr 3451  EXMIDwem 4035  1oc1o 6188  2oc2o 6189 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-nul 3971 This theorem depends on definitions:  df-bi 116  df-dc 782  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-sn 3456  df-pr 3457  df-exmid 4036  df-suc 4207  df-1o 6195  df-2o 6196 This theorem is referenced by: (None)
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