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Theorem ss1oel2o 13219
Description: Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4121 which more directly illustrates the contrast with el2oss1o 13218. (Contributed by Jim Kingdon, 8-Aug-2022.)
Assertion
Ref Expression
ss1oel2o (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o𝑥 ∈ 2o))

Proof of Theorem ss1oel2o
StepHypRef Expression
1 exmid01 4121 . 2 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2 df1o2 6326 . . . . 5 1o = {∅}
32sseq2i 3124 . . . 4 (𝑥 ⊆ 1o𝑥 ⊆ {∅})
4 df2o2 6328 . . . . . 6 2o = {∅, {∅}}
54eleq2i 2206 . . . . 5 (𝑥 ∈ 2o𝑥 ∈ {∅, {∅}})
6 vex 2689 . . . . . 6 𝑥 ∈ V
76elpr 3548 . . . . 5 (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
85, 7bitri 183 . . . 4 (𝑥 ∈ 2o ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
93, 8imbi12i 238 . . 3 ((𝑥 ⊆ 1o𝑥 ∈ 2o) ↔ (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
109albii 1446 . 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 697  wal 1329   = wceq 1331  wcel 1480  wss 3071  c0 3363  {csn 3527  {cpr 3528  EXMIDwem 4118  1oc1o 6306  2oc2o 6307
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054
This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-exmid 4119  df-suc 4293  df-1o 6313  df-2o 6314
This theorem is referenced by: (None)
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