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Theorem onsucelsucexmidlem 4281
Description: Lemma for onsucelsucexmid 4282. The set {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} appears as 𝐴 in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5530), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4272. (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 489 . . . . . . . 8 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) ∧ 𝑧 = ∅) → 𝑦𝑧)
2 noel 3255 . . . . . . . . . 10 ¬ 𝑦 ∈ ∅
3 eleq2 2117 . . . . . . . . . 10 (𝑧 = ∅ → (𝑦𝑧𝑦 ∈ ∅))
42, 3mtbiri 610 . . . . . . . . 9 (𝑧 = ∅ → ¬ 𝑦𝑧)
54adantl 266 . . . . . . . 8 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) ∧ 𝑧 = ∅) → ¬ 𝑦𝑧)
61, 5pm2.21dd 560 . . . . . . 7 (((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) ∧ 𝑧 = ∅) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
76ex 112 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → (𝑧 = ∅ → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}))
8 eleq2 2117 . . . . . . . . . . 11 (𝑧 = {∅} → (𝑦𝑧𝑦 ∈ {∅}))
98biimpac 286 . . . . . . . . . 10 ((𝑦𝑧𝑧 = {∅}) → 𝑦 ∈ {∅})
10 velsn 3419 . . . . . . . . . 10 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
119, 10sylib 131 . . . . . . . . 9 ((𝑦𝑧𝑧 = {∅}) → 𝑦 = ∅)
12 onsucelsucexmidlem1 4280 . . . . . . . . 9 ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
1311, 12syl6eqel 2144 . . . . . . . 8 ((𝑦𝑧𝑧 = {∅}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
1413ex 112 . . . . . . 7 (𝑦𝑧 → (𝑧 = {∅} → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}))
1514adantr 265 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → (𝑧 = {∅} → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}))
16 elrabi 2717 . . . . . . . 8 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} → 𝑧 ∈ {∅, {∅}})
17 vex 2577 . . . . . . . . 9 𝑧 ∈ V
1817elpr 3423 . . . . . . . 8 (𝑧 ∈ {∅, {∅}} ↔ (𝑧 = ∅ ∨ 𝑧 = {∅}))
1916, 18sylib 131 . . . . . . 7 (𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} → (𝑧 = ∅ ∨ 𝑧 = {∅}))
2019adantl 266 . . . . . 6 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → (𝑧 = ∅ ∨ 𝑧 = {∅}))
217, 15, 20mpjaod 648 . . . . 5 ((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
2221gen2 1355 . . . 4 𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
23 dftr2 3883 . . . 4 (Tr {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}) → 𝑦 ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}))
2422, 23mpbir 138 . . 3 Tr {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
25 ssrab2 3052 . . 3 {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ⊆ {∅, {∅}}
26 2ordpr 4276 . . 3 Ord {∅, {∅}}
27 trssord 4144 . . 3 ((Tr {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∧ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ⊆ {∅, {∅}} ∧ Ord {∅, {∅}}) → Ord {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
2824, 25, 26, 27mp3an 1243 . 2 Ord {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
29 pp0ex 3967 . . . 4 {∅, {∅}} ∈ V
3029rabex 3928 . . 3 {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ V
3130elon 4138 . 2 ({𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On ↔ Ord {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
3228, 31mpbir 138 1 {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wo 639  wal 1257   = wceq 1259  wcel 1409  {crab 2327  wss 2944  c0 3251  {csn 3402  {cpr 3403  Tr wtr 3881  Ord word 4126  Oncon0 4127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-nul 3910  ax-pow 3954
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-uni 3608  df-tr 3882  df-iord 4130  df-on 4132  df-suc 4135
This theorem is referenced by:  onsucelsucexmid  4282  acexmidlemcase  5534  acexmidlemv  5537
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