Theorem tz9.13 9480

Description: Every set is well-founded, assuming the Axiom of Regularity. In other words, every set belongs to a layer of the cumulative hierarchy of sets. Proposition 9.13 of [TakeutiZaring] p. 78. (Contributed by NM, 23-Sep-2003.)

Hypothesis

Ref	Expression
tz9.13.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
tz9.13	⊢ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)

Distinct variable group:   𝑥,𝐴

Proof of Theorem tz9.13

Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tz9.13.1	. . 3 ⊢ 𝐴 ∈ V
2		setind 9423	. . . 4 ⊢ (∀𝑧(𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)}) → {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} = V)
3		ssel 3910	. . . . . . . 8 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → (𝑤 ∈ 𝑧 → 𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)}))
4		vex 3426	. . . . . . . . 9 ⊢ 𝑤 ∈ V
5		eleq1 2826	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝑅1‘𝑥) ↔ 𝑤 ∈ (𝑅1‘𝑥)))
6	5	rexbidv 3225	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥)))
7	4, 6	elab 3602	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥))
8	3, 7	syl6ib 250	. . . . . . 7 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → (𝑤 ∈ 𝑧 → ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥)))
9	8	ralrimiv 3106	. . . . . 6 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → ∀𝑤 ∈ 𝑧 ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥))
10		vex 3426	. . . . . . 7 ⊢ 𝑧 ∈ V
11	10	tz9.12 9479	. . . . . 6 ⊢ (∀𝑤 ∈ 𝑧 ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥) → ∃𝑥 ∈ On 𝑧 ∈ (𝑅1‘𝑥))
12	9, 11	syl 17	. . . . 5 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → ∃𝑥 ∈ On 𝑧 ∈ (𝑅1‘𝑥))
13		eleq1 2826	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (𝑅1‘𝑥) ↔ 𝑧 ∈ (𝑅1‘𝑥)))
14	13	rexbidv 3225	. . . . . 6 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1‘𝑥)))
15	10, 14	elab 3602	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1‘𝑥))
16	12, 15	sylibr 233	. . . 4 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)})
17	2, 16	mpg 1801	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} = V
18	1, 17	eleqtrri 2838	. 2 ⊢ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)}
19		eleq1 2826	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1‘𝑥) ↔ 𝐴 ∈ (𝑅1‘𝑥)))
20	19	rexbidv 3225	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)))
21	1, 20	elab 3602	. 2 ⊢ (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥))
22	18, 21	mpbi 229	1 ⊢ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  Oncon0 6251  ‘cfv 6418  𝑅₁cr1 9451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-reg 9281  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-r1 9453

This theorem is referenced by:  tz9.13g  9481  elhf2  34404