Theorem infeq5 9100

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 9106.) 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.

Step	Hyp	Ref	Expression
1		df-pss 3954	. . . . 5 ⊢ (𝑥 ⊊ ∪ 𝑥 ↔ (𝑥 ⊆ ∪ 𝑥 ∧ 𝑥 ≠ ∪ 𝑥))
2		unieq 4849	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
3		uni0 4866	. . . . . . . . . 10 ⊢ ∪ ∅ = ∅
4	2, 3	syl6req 2873	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ∅ = ∪ 𝑥)
5		eqtr 2841	. . . . . . . . 9 ⊢ ((𝑥 = ∅ ∧ ∅ = ∪ 𝑥) → 𝑥 = ∪ 𝑥)
6	4, 5	mpdan 685	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝑥 = ∪ 𝑥)
7	6	necon3i 3048	. . . . . . 7 ⊢ (𝑥 ≠ ∪ 𝑥 → 𝑥 ≠ ∅)
8	7	anim1i 616	. . . . . 6 ⊢ ((𝑥 ≠ ∪ 𝑥 ∧ 𝑥 ⊆ ∪ 𝑥) → (𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥))
9	8	ancoms 461	. . . . 5 ⊢ ((𝑥 ⊆ ∪ 𝑥 ∧ 𝑥 ≠ ∪ 𝑥) → (𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥))
10	1, 9	sylbi 219	. . . 4 ⊢ (𝑥 ⊊ ∪ 𝑥 → (𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥))
11	10	eximi 1835	. . 3 ⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 → ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥))
12		eqid 2821	. . . . 5 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})
13		eqid 2821	. . . . 5 ⊢ (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω) = (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω)
14		vex 3497	. . . . 5 ⊢ 𝑥 ∈ V
15	12, 13, 14, 14	inf3lem7 9097	. . . 4 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ∈ V)
16	15	exlimiv 1931	. . 3 ⊢ (∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ∈ V)
17	11, 16	syl 17	. 2 ⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 → ω ∈ V)
18		infeq5i 9099	. 2 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥)
19	17, 18	impbii 211	1 ⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 ↔ ω ∈ V)

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3016  {crab 3142  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936   ⊊ wpss 3937  ∅c0 4291  ∪ cuni 4838   ↦ cmpt 5146   ↾ cres 5557  ωcom 7580  reccrdg 8045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-reg 9056

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046

This theorem is referenced by: (None)