Theorem tz9.13 9706

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 9659	. . . 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 3909	. . . . . . . 8 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → (𝑤 ∈ 𝑧 → 𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)}))
4		vex 3435	. . . . . . . . 9 ⊢ 𝑤 ∈ V
5		eleq1 2827	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝑅1‘𝑥) ↔ 𝑤 ∈ (𝑅1‘𝑥)))
6	5	rexbidv 3163	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥)))
7	4, 6	elab 3617	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥))
8	3, 7	imbitrdi 252	. . . . . . 7 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → (𝑤 ∈ 𝑧 → ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥)))
9	8	ralrimiv 3130	. . . . . 6 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → ∀𝑤 ∈ 𝑧 ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥))
10		vex 3435	. . . . . . 7 ⊢ 𝑧 ∈ V
11	10	tz9.12 9705	. . . . . 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 2827	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (𝑅1‘𝑥) ↔ 𝑧 ∈ (𝑅1‘𝑥)))
14	13	rexbidv 3163	. . . . . 6 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1‘𝑥)))
15	10, 14	elab 3617	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1‘𝑥))
16	12, 15	sylibr 235	. . . 4 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)})
17	2, 16	mpg 1804	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} = V
18	1, 17	eleqtrri 2838	. 2 ⊢ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)}
19		eleq1 2827	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1‘𝑥) ↔ 𝐴 ∈ (𝑅1‘𝑥)))
20	19	rexbidv 3163	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)))
21	1, 20	elab 3617	. 2 ⊢ (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥))
22	18, 21	mpbi 231	1 ⊢ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  {cab 2717  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883  Oncon0 6310  ‘cfv 6485  𝑅₁cr1 9677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-reg 9497  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-r1 9679

This theorem is referenced by:  tz9.13g  9707  elhf2  36403