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Theorem ordtri2or2exmidlem 4537
Description: A set which is 2o if 𝜑 or if ¬ 𝜑 is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)
Assertion
Ref Expression
ordtri2or2exmidlem {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
Distinct variable group:   𝜑,𝑥

Proof of Theorem ordtri2or2exmidlem
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 527 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → 𝑦𝑧)
2 noel 3438 . . . . . . . . 9 ¬ 𝑦 ∈ ∅
3 eleq2 2251 . . . . . . . . 9 (𝑧 = ∅ → (𝑦𝑧𝑦 ∈ ∅))
42, 3mtbiri 676 . . . . . . . 8 (𝑧 = ∅ → ¬ 𝑦𝑧)
54adantl 277 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → ¬ 𝑦𝑧)
61, 5pm2.21dd 621 . . . . . 6 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = ∅) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
7 eleq2 2251 . . . . . . . . . . 11 (𝑧 = {∅} → (𝑦𝑧𝑦 ∈ {∅}))
87biimpac 298 . . . . . . . . . 10 ((𝑦𝑧𝑧 = {∅}) → 𝑦 ∈ {∅})
9 velsn 3621 . . . . . . . . . 10 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
108, 9sylib 122 . . . . . . . . 9 ((𝑦𝑧𝑧 = {∅}) → 𝑦 = ∅)
11 orc 713 . . . . . . . . . 10 (𝑦 = ∅ → (𝑦 = ∅ ∨ 𝑦 = {∅}))
12 vex 2752 . . . . . . . . . . 11 𝑦 ∈ V
1312elpr 3625 . . . . . . . . . 10 (𝑦 ∈ {∅, {∅}} ↔ (𝑦 = ∅ ∨ 𝑦 = {∅}))
1411, 13sylibr 134 . . . . . . . . 9 (𝑦 = ∅ → 𝑦 ∈ {∅, {∅}})
1510, 14syl 14 . . . . . . . 8 ((𝑦𝑧𝑧 = {∅}) → 𝑦 ∈ {∅, {∅}})
1615adantlr 477 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝑦 ∈ {∅, {∅}})
17 biidd 172 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝜑𝜑))
1817elrab 2905 . . . . . . . . 9 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ (𝑧 ∈ {∅, {∅}} ∧ 𝜑))
1918simprbi 275 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → 𝜑)
2019ad2antlr 489 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝜑)
21 biidd 172 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜑))
2221elrab 2905 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ (𝑦 ∈ {∅, {∅}} ∧ 𝜑))
2316, 20, 22sylanbrc 417 . . . . . 6 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) ∧ 𝑧 = {∅}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
24 elrabi 2902 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → 𝑧 ∈ {∅, {∅}})
25 vex 2752 . . . . . . . . 9 𝑧 ∈ V
2625elpr 3625 . . . . . . . 8 (𝑧 ∈ {∅, {∅}} ↔ (𝑧 = ∅ ∨ 𝑧 = {∅}))
2724, 26sylib 122 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} → (𝑧 = ∅ ∨ 𝑧 = {∅}))
2827adantl 277 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → (𝑧 = ∅ ∨ 𝑧 = {∅}))
296, 23, 28mpjaodan 799 . . . . 5 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
3029gen2 1460 . . . 4 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
31 dftr2 4115 . . . 4 (Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ 𝜑}))
3230, 31mpbir 146 . . 3 Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑}
33 ssrab2 3252 . . 3 {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅, {∅}}
34 2ordpr 4535 . . 3 Ord {∅, {∅}}
35 trssord 4392 . . 3 ((Tr {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∧ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ⊆ {∅, {∅}} ∧ Ord {∅, {∅}}) → Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
3632, 33, 34, 35mp3an 1347 . 2 Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑}
37 pp0ex 4201 . . . 4 {∅, {∅}} ∈ V
3837rabex 4159 . . 3 {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ V
3938elon 4386 . 2 ({𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On ↔ Ord {𝑥 ∈ {∅, {∅}} ∣ 𝜑})
4036, 39mpbir 146 1 {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 709  wal 1361   = wceq 1363  wcel 2158  {crab 2469  wss 3141  c0 3434  {csn 3604  {cpr 3605  Tr wtr 4113  Ord word 4374  Oncon0 4375
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380  df-suc 4383
This theorem is referenced by:  ordtri2or2exmid  4582  ontri2orexmidim  4583
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