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Theorem infeq5 8709
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 8715.) 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 3731 . . . . 5 (𝑥 𝑥 ↔ (𝑥 𝑥𝑥 𝑥))
2 unieq 4596 . . . . . . . . . 10 (𝑥 = ∅ → 𝑥 = ∅)
3 uni0 4617 . . . . . . . . . 10 ∅ = ∅
42, 3syl6req 2811 . . . . . . . . 9 (𝑥 = ∅ → ∅ = 𝑥)
5 eqtr 2779 . . . . . . . . 9 ((𝑥 = ∅ ∧ ∅ = 𝑥) → 𝑥 = 𝑥)
64, 5mpdan 705 . . . . . . . 8 (𝑥 = ∅ → 𝑥 = 𝑥)
76necon3i 2964 . . . . . . 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 1911 . . 3 (∃𝑥 𝑥 𝑥 → ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥))
12 eqid 2760 . . . . 5 (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
13 eqid 2760 . . . . 5 (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω) = (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω)
14 vex 3343 . . . . 5 𝑥 ∈ V
1512, 13, 14, 14inf3lem7 8706 . . . 4 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ∈ V)
1615exlimiv 2007 . . 3 (∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ∈ V)
1711, 16syl 17 . 2 (∃𝑥 𝑥 𝑥 → ω ∈ V)
18 infeq5i 8708 . 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 1632  wex 1853  wcel 2139  wne 2932  {crab 3054  Vcvv 3340  cin 3714  wss 3715  wpss 3716  c0 4058   cuni 4588  cmpt 4881  cres 5268  ωcom 7231  reccrdg 7675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-reg 8664
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676
This theorem is referenced by: (None)
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