Theorem infeq5 9597

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 9603.) 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 3937	. . . . 5 ⊢ (𝑥 ⊊ ∪ 𝑥 ↔ (𝑥 ⊆ ∪ 𝑥 ∧ 𝑥 ≠ ∪ 𝑥))
2		unieq 4885	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
3		uni0 4902	. . . . . . . . . 10 ⊢ ∪ ∅ = ∅
4	2, 3	eqtr2di 2782	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ∅ = ∪ 𝑥)
5		eqtr 2750	. . . . . . . . 9 ⊢ ((𝑥 = ∅ ∧ ∅ = ∪ 𝑥) → 𝑥 = ∪ 𝑥)
6	4, 5	mpdan 687	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝑥 = ∪ 𝑥)
7	6	necon3i 2958	. . . . . . 7 ⊢ (𝑥 ≠ ∪ 𝑥 → 𝑥 ≠ ∅)
8	7	anim1i 615	. . . . . 6 ⊢ ((𝑥 ≠ ∪ 𝑥 ∧ 𝑥 ⊆ ∪ 𝑥) → (𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥))
9	8	ancoms 458	. . . . 5 ⊢ ((𝑥 ⊆ ∪ 𝑥 ∧ 𝑥 ≠ ∪ 𝑥) → (𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥))
10	1, 9	sylbi 217	. . . 4 ⊢ (𝑥 ⊊ ∪ 𝑥 → (𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥))
11	10	eximi 1835	. . 3 ⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 → ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥))
12		eqid 2730	. . . . 5 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})
13		eqid 2730	. . . . 5 ⊢ (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω) = (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω)
14		vex 3454	. . . . 5 ⊢ 𝑥 ∈ V
15	12, 13, 14, 14	inf3lem7 9594	. . . 4 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ∈ V)
16	15	exlimiv 1930	. . 3 ⊢ (∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ∈ V)
17	11, 16	syl 17	. 2 ⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 → ω ∈ V)
18		infeq5i 9596	. 2 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥)
19	17, 18	impbii 209	1 ⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 ↔ ω ∈ V)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  {crab 3408  Vcvv 3450   ∩ cin 3916   ⊆ wss 3917   ⊊ wpss 3918  ∅c0 4299  ∪ cuni 4874   ↦ cmpt 5191   ↾ cres 5643  ωcom 7845  reccrdg 8380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-reg 9552

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381

This theorem is referenced by: (None)