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Theorem infeq5 8753
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 8759.) 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 3750 . . . . 5 (𝑥 𝑥 ↔ (𝑥 𝑥𝑥 𝑥))
2 unieq 4604 . . . . . . . . . 10 (𝑥 = ∅ → 𝑥 = ∅)
3 uni0 4625 . . . . . . . . . 10 ∅ = ∅
42, 3syl6req 2816 . . . . . . . . 9 (𝑥 = ∅ → ∅ = 𝑥)
5 eqtr 2784 . . . . . . . . 9 ((𝑥 = ∅ ∧ ∅ = 𝑥) → 𝑥 = 𝑥)
64, 5mpdan 678 . . . . . . . 8 (𝑥 = ∅ → 𝑥 = 𝑥)
76necon3i 2969 . . . . . . 7 (𝑥 𝑥𝑥 ≠ ∅)
87anim1i 608 . . . . . 6 ((𝑥 𝑥𝑥 𝑥) → (𝑥 ≠ ∅ ∧ 𝑥 𝑥))
98ancoms 450 . . . . 5 ((𝑥 𝑥𝑥 𝑥) → (𝑥 ≠ ∅ ∧ 𝑥 𝑥))
101, 9sylbi 208 . . . 4 (𝑥 𝑥 → (𝑥 ≠ ∅ ∧ 𝑥 𝑥))
1110eximi 1929 . . 3 (∃𝑥 𝑥 𝑥 → ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥))
12 eqid 2765 . . . . 5 (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
13 eqid 2765 . . . . 5 (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω) = (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω)
14 vex 3353 . . . . 5 𝑥 ∈ V
1512, 13, 14, 14inf3lem7 8750 . . . 4 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ∈ V)
1615exlimiv 2025 . . 3 (∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ∈ V)
1711, 16syl 17 . 2 (∃𝑥 𝑥 𝑥 → ω ∈ V)
18 infeq5i 8752 . 2 (ω ∈ V → ∃𝑥 𝑥 𝑥)
1917, 18impbii 200 1 (∃𝑥 𝑥 𝑥 ↔ ω ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wne 2937  {crab 3059  Vcvv 3350  cin 3733  wss 3734  wpss 3735  c0 4081   cuni 4596  cmpt 4890  cres 5281  ωcom 7267  reccrdg 7713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-reg 8708
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-om 7268  df-wrecs 7614  df-recs 7676  df-rdg 7714
This theorem is referenced by: (None)
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