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Theorem onsucelsucexmidlem 4335
Description: Lemma for onsucelsucexmid 4336. The set {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} appears as 𝐴 in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5625), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4326. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
onsucelsucexmidlem {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On
Distinct variable group:   𝜑,𝑥

Proof of Theorem onsucelsucexmidlem
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 496 . . . . . . . 8 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) ∧ 𝑧 = ∅) → 𝑦𝑧)
2 noel 3288 . . . . . . . . . 10 ¬ 𝑦 ∈ ∅
3 eleq2 2151 . . . . . . . . . 10 (𝑧 = ∅ → (𝑦𝑧𝑦 ∈ ∅))
42, 3mtbiri 635 . . . . . . . . 9 (𝑧 = ∅ → ¬ 𝑦𝑧)
54adantl 271 . . . . . . . 8 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) ∧ 𝑧 = ∅) → ¬ 𝑦𝑧)
61, 5pm2.21dd 585 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) ∧ 𝑧 = ∅) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
76ex 113 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → (𝑧 = ∅ → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}))
8 eleq2 2151 . . . . . . . . . . 11 (𝑧 = {∅} → (𝑦𝑧𝑦 ∈ {∅}))
98biimpac 292 . . . . . . . . . 10 ((𝑦𝑧𝑧 = {∅}) → 𝑦 ∈ {∅})
10 velsn 3458 . . . . . . . . . 10 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
119, 10sylib 120 . . . . . . . . 9 ((𝑦𝑧𝑧 = {∅}) → 𝑦 = ∅)
12 onsucelsucexmidlem1 4334 . . . . . . . . 9 ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
1311, 12syl6eqel 2178 . . . . . . . 8 ((𝑦𝑧𝑧 = {∅}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
1413ex 113 . . . . . . 7 (𝑦𝑧 → (𝑧 = {∅} → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}))
1514adantr 270 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → (𝑧 = {∅} → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}))
16 elrabi 2766 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} → 𝑧 ∈ {∅, {∅}})
17 vex 2622 . . . . . . . . 9 𝑧 ∈ V
1817elpr 3462 . . . . . . . 8 (𝑧 ∈ {∅, {∅}} ↔ (𝑧 = ∅ ∨ 𝑧 = {∅}))
1916, 18sylib 120 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} → (𝑧 = ∅ ∨ 𝑧 = {∅}))
2019adantl 271 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → (𝑧 = ∅ ∨ 𝑧 = {∅}))
217, 15, 20mpjaod 673 . . . . 5 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
2221gen2 1384 . . . 4 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
23 dftr2 3930 . . . 4 (Tr {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}))
2422, 23mpbir 144 . . 3 Tr {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
25 ssrab2 3104 . . 3 {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ⊆ {∅, {∅}}
26 2ordpr 4330 . . 3 Ord {∅, {∅}}
27 trssord 4198 . . 3 ((Tr {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∧ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ⊆ {∅, {∅}} ∧ Ord {∅, {∅}}) → Ord {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
2824, 25, 26, 27mp3an 1273 . 2 Ord {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
29 pp0ex 4015 . . . 4 {∅, {∅}} ∈ V
3029rabex 3975 . . 3 {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ V
3130elon 4192 . 2 ({𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On ↔ Ord {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
3228, 31mpbir 144 1 {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 664  wal 1287   = wceq 1289  wcel 1438  {crab 2363  wss 2997  c0 3284  {csn 3441  {cpr 3442  Tr wtr 3928  Ord word 4180  Oncon0 4181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-tr 3929  df-iord 4184  df-on 4186  df-suc 4189
This theorem is referenced by:  onsucelsucexmid  4336  acexmidlemcase  5629  acexmidlemv  5632
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