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Theorem infeq5 8478
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 8484.) 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 3571 . . . . 5 (𝑥 𝑥 ↔ (𝑥 𝑥𝑥 𝑥))
2 unieq 4410 . . . . . . . . . 10 (𝑥 = ∅ → 𝑥 = ∅)
3 uni0 4431 . . . . . . . . . 10 ∅ = ∅
42, 3syl6req 2672 . . . . . . . . 9 (𝑥 = ∅ → ∅ = 𝑥)
5 eqtr 2640 . . . . . . . . 9 ((𝑥 = ∅ ∧ ∅ = 𝑥) → 𝑥 = 𝑥)
64, 5mpdan 701 . . . . . . . 8 (𝑥 = ∅ → 𝑥 = 𝑥)
76necon3i 2822 . . . . . . 7 (𝑥 𝑥𝑥 ≠ ∅)
87anim1i 591 . . . . . 6 ((𝑥 𝑥𝑥 𝑥) → (𝑥 ≠ ∅ ∧ 𝑥 𝑥))
98ancoms 469 . . . . 5 ((𝑥 𝑥𝑥 𝑥) → (𝑥 ≠ ∅ ∧ 𝑥 𝑥))
101, 9sylbi 207 . . . 4 (𝑥 𝑥 → (𝑥 ≠ ∅ ∧ 𝑥 𝑥))
1110eximi 1759 . . 3 (∃𝑥 𝑥 𝑥 → ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥))
12 eqid 2621 . . . . 5 (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
13 eqid 2621 . . . . 5 (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω) = (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω)
14 vex 3189 . . . . 5 𝑥 ∈ V
1512, 13, 14, 14inf3lem7 8475 . . . 4 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ∈ V)
1615exlimiv 1855 . . 3 (∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ∈ V)
1711, 16syl 17 . 2 (∃𝑥 𝑥 𝑥 → ω ∈ V)
18 infeq5i 8477 . 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 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  {crab 2911  Vcvv 3186  cin 3554  wss 3555  wpss 3556  c0 3891   cuni 4402  cmpt 4673  cres 5076  ωcom 7012  reccrdg 7450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-reg 8441
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451
This theorem is referenced by: (None)
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