Theorem tz9.13 8654

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 8610	. . . 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 3597	. . . . . . . 8 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → (𝑤 ∈ 𝑧 → 𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)}))
4		vex 3203	. . . . . . . . 9 ⊢ 𝑤 ∈ V
5		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝑅1‘𝑥) ↔ 𝑤 ∈ (𝑅1‘𝑥)))
6	5	rexbidv 3052	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥)))
7	4, 6	elab 3350	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥))
8	3, 7	syl6ib 241	. . . . . . 7 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → (𝑤 ∈ 𝑧 → ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥)))
9	8	ralrimiv 2965	. . . . . 6 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → ∀𝑤 ∈ 𝑧 ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥))
10		vex 3203	. . . . . . 7 ⊢ 𝑧 ∈ V
11	10	tz9.12 8653	. . . . . 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 2689	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (𝑅1‘𝑥) ↔ 𝑧 ∈ (𝑅1‘𝑥)))
14	13	rexbidv 3052	. . . . . 6 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1‘𝑥)))
15	10, 14	elab 3350	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1‘𝑥))
16	12, 15	sylibr 224	. . . 4 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)})
17	2, 16	mpg 1724	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} = V
18	1, 17	eleqtrri 2700	. 2 ⊢ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)}
19		eleq1 2689	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1‘𝑥) ↔ 𝐴 ∈ (𝑅1‘𝑥)))
20	19	rexbidv 3052	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)))
21	1, 20	elab 3350	. 2 ⊢ (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥))
22	18, 21	mpbi 220	1 ⊢ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  Oncon0 5723  ‘cfv 5888  𝑅₁cr1 8625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627

This theorem is referenced by:  tz9.13g  8655  elhf2  32282