Theorem tz9.13 9214

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 9170	. . . 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 3961	. . . . . . . 8 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → (𝑤 ∈ 𝑧 → 𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)}))
4		vex 3498	. . . . . . . . 9 ⊢ 𝑤 ∈ V
5		eleq1 2900	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝑅1‘𝑥) ↔ 𝑤 ∈ (𝑅1‘𝑥)))
6	5	rexbidv 3297	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥)))
7	4, 6	elab 3667	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥))
8	3, 7	syl6ib 253	. . . . . . 7 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → (𝑤 ∈ 𝑧 → ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥)))
9	8	ralrimiv 3181	. . . . . 6 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → ∀𝑤 ∈ 𝑧 ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥))
10		vex 3498	. . . . . . 7 ⊢ 𝑧 ∈ V
11	10	tz9.12 9213	. . . . . 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 2900	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (𝑅1‘𝑥) ↔ 𝑧 ∈ (𝑅1‘𝑥)))
14	13	rexbidv 3297	. . . . . 6 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1‘𝑥)))
15	10, 14	elab 3667	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1‘𝑥))
16	12, 15	sylibr 236	. . . 4 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)})
17	2, 16	mpg 1794	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} = V
18	1, 17	eleqtrri 2912	. 2 ⊢ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)}
19		eleq1 2900	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1‘𝑥) ↔ 𝐴 ∈ (𝑅1‘𝑥)))
20	19	rexbidv 3297	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)))
21	1, 20	elab 3667	. 2 ⊢ (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥))
22	18, 21	mpbi 232	1 ⊢ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  {cab 2799  ∀wral 3138  ∃wrex 3139  Vcvv 3495   ⊆ wss 3936  Oncon0 6186  ‘cfv 6350  𝑅₁cr1 9185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-reg 9050  ax-inf2 9098

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-r1 9187

This theorem is referenced by:  tz9.13g  9215  elhf2  33631