MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  infeq5 Structured version   Visualization version   GIF version

Theorem infeq5 8703
Description: The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (shown on the right-hand side in the form of omex 8709.) The left-hand side provides us with a very short way to express the Axiom of Infinity using only elementary symbols. This proof of equivalence does not depend on the Axiom of Infinity. (Contributed by NM, 23-Mar-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
infeq5 (∃𝑥 𝑥 𝑥 ↔ ω ∈ V)

Proof of Theorem infeq5
Dummy variables 𝑦 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pss 3727 . . . . 5 (𝑥 𝑥 ↔ (𝑥 𝑥𝑥 𝑥))
2 unieq 4592 . . . . . . . . . 10 (𝑥 = ∅ → 𝑥 = ∅)
3 uni0 4613 . . . . . . . . . 10 ∅ = ∅
42, 3syl6req 2807 . . . . . . . . 9 (𝑥 = ∅ → ∅ = 𝑥)
5 eqtr 2775 . . . . . . . . 9 ((𝑥 = ∅ ∧ ∅ = 𝑥) → 𝑥 = 𝑥)
64, 5mpdan 705 . . . . . . . 8 (𝑥 = ∅ → 𝑥 = 𝑥)
76necon3i 2960 . . . . . . 7 (𝑥 𝑥𝑥 ≠ ∅)
87anim1i 593 . . . . . 6 ((𝑥 𝑥𝑥 𝑥) → (𝑥 ≠ ∅ ∧ 𝑥 𝑥))
98ancoms 468 . . . . 5 ((𝑥 𝑥𝑥 𝑥) → (𝑥 ≠ ∅ ∧ 𝑥 𝑥))
101, 9sylbi 207 . . . 4 (𝑥 𝑥 → (𝑥 ≠ ∅ ∧ 𝑥 𝑥))
1110eximi 1907 . . 3 (∃𝑥 𝑥 𝑥 → ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥))
12 eqid 2756 . . . . 5 (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
13 eqid 2756 . . . . 5 (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω) = (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω)
14 vex 3339 . . . . 5 𝑥 ∈ V
1512, 13, 14, 14inf3lem7 8700 . . . 4 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ∈ V)
1615exlimiv 2003 . . 3 (∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ∈ V)
1711, 16syl 17 . 2 (∃𝑥 𝑥 𝑥 → ω ∈ V)
18 infeq5i 8702 . 2 (ω ∈ V → ∃𝑥 𝑥 𝑥)
1917, 18impbii 199 1 (∃𝑥 𝑥 𝑥 ↔ ω ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1628  wex 1849  wcel 2135  wne 2928  {crab 3050  Vcvv 3336  cin 3710  wss 3711  wpss 3712  c0 4054   cuni 4584  cmpt 4877  cres 5264  ωcom 7226  reccrdg 7670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-reg 8658
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-om 7227  df-wrecs 7572  df-recs 7633  df-rdg 7671
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator