Theorem infeq5 9546

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 9552.) 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 3921	. . . . 5 ⊢ (𝑥 ⊊ ∪ 𝑥 ↔ (𝑥 ⊆ ∪ 𝑥 ∧ 𝑥 ≠ ∪ 𝑥))
2		unieq 4874	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
3		uni0 4891	. . . . . . . . . 10 ⊢ ∪ ∅ = ∅
4	2, 3	eqtr2di 2788	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ∅ = ∪ 𝑥)
5		eqtr 2756	. . . . . . . . 9 ⊢ ((𝑥 = ∅ ∧ ∅ = ∪ 𝑥) → 𝑥 = ∪ 𝑥)
6	4, 5	mpdan 687	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝑥 = ∪ 𝑥)
7	6	necon3i 2964	. . . . . . 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 1836	. . 3 ⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 → ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥))
12		eqid 2736	. . . . 5 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})
13		eqid 2736	. . . . 5 ⊢ (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω) = (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω)
14		vex 3444	. . . . 5 ⊢ 𝑥 ∈ V
15	12, 13, 14, 14	inf3lem7 9543	. . . 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 9545	. 2 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥)
19	17, 18	impbii 209	1 ⊢ (∃𝑥 𝑥 ⊊ ∪ 𝑥 ↔ ω ∈ V)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  {crab 3399  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901   ⊊ wpss 3902  ∅c0 4285  ∪ cuni 4863   ↦ cmpt 5179   ↾ cres 5626  ωcom 7808  reccrdg 8340

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-reg 9497

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341

This theorem is referenced by: (None)