Theorem infeq5 8417

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 8423.) 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 3556	. . . . 5 ⊢ (𝑥 ⊊ ∪ 𝑥 ↔ (𝑥 ⊆ ∪ 𝑥 ∧ 𝑥 ≠ ∪ 𝑥))
2		unieq 4380	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
3		uni0 4401	. . . . . . . . . 10 ⊢ ∪ ∅ = ∅
4	2, 3	syl6req 2661	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ∅ = ∪ 𝑥)
5		eqtr 2629	. . . . . . . . 9 ⊢ ((𝑥 = ∅ ∧ ∅ = ∪ 𝑥) → 𝑥 = ∪ 𝑥)
6	4, 5	mpdan 699	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝑥 = ∪ 𝑥)
7	6	necon3i 2814	. . . . . . 7 ⊢ (𝑥 ≠ ∪ 𝑥 → 𝑥 ≠ ∅)
8	7	anim1i 590	. . . . . 6 ⊢ ((𝑥 ≠ ∪ 𝑥 ∧ 𝑥 ⊆ ∪ 𝑥) → (𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥))
9	8	ancoms 468	. . . . 5 ⊢ ((𝑥 ⊆ ∪ 𝑥 ∧ 𝑥 ≠ ∪ 𝑥) → (𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥))
10	1, 9	sylbi 206	. . . 4 ⊢ (𝑥 ⊊ ∪ 𝑥 → (𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥))
11	10	eximi 1752	. . 3 ⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 → ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥))
12		eqid 2610	. . . . 5 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})
13		eqid 2610	. . . . 5 ⊢ (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω) = (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω)
14		vex 3176	. . . . 5 ⊢ 𝑥 ∈ V
15	12, 13, 14, 14	inf3lem7 8414	. . . 4 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ∈ V)
16	15	exlimiv 1845	. . 3 ⊢ (∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ∈ V)
17	11, 16	syl 17	. 2 ⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 → ω ∈ V)
18		infeq5i 8416	. 2 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥)
19	17, 18	impbii 198	1 ⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 ↔ ω ∈ V)

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   ⊊ wpss 3541  ∅c0 3874  ∪ cuni 4372   ↦ cmpt 4643   ↾ cres 5040  ωcom 6957  reccrdg 7392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393

This theorem is referenced by: (None)