Theorem rankwflemb 9824

Description: Two ways of expressing that a set is well-founded. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.) (Proof shortened by BJ, 29-Sep-2026.)

Assertion

Ref	Expression
rankwflemb	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem rankwflemb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 4873	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ (𝑅1 “ On)))
2		eleq2 2853	. . . . . . . 8 ⊢ ((𝑅1‘𝑥) = 𝑦 → (𝐴 ∈ (𝑅1‘𝑥) ↔ 𝐴 ∈ 𝑦))
3	2	biimprcd 253	. . . . . . 7 ⊢ (𝐴 ∈ 𝑦 → ((𝑅1‘𝑥) = 𝑦 → 𝐴 ∈ (𝑅1‘𝑥)))
4		r1tr 9807	. . . . . . . . . . 11 ⊢ Tr (𝑅1‘𝑥)
5		trss 5225	. . . . . . . . . . 11 ⊢ (Tr (𝑅1‘𝑥) → (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ⊆ (𝑅1‘𝑥)))
6	4, 5	ax-mp 5	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ⊆ (𝑅1‘𝑥))
7		elpwg 4563	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → (𝐴 ∈ 𝒫 (𝑅1‘𝑥) ↔ 𝐴 ⊆ (𝑅1‘𝑥)))
8	6, 7	mpbird 260	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ∈ 𝒫 (𝑅1‘𝑥))
9		elfvdm 6939	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → 𝑥 ∈ dom 𝑅1)
10		r1sucg 9800	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
11	9, 10	syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
12	8, 11	eleqtrrd 2867	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ∈ (𝑅1‘suc 𝑥))
13	12	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ On → (𝐴 ∈ (𝑅1‘𝑥) → 𝐴 ∈ (𝑅1‘suc 𝑥)))
14	3, 13	syl9 78	. . . . . 6 ⊢ (𝐴 ∈ 𝑦 → (𝑥 ∈ On → ((𝑅1‘𝑥) = 𝑦 → 𝐴 ∈ (𝑅1‘suc 𝑥))))
15	14	reximdvai 3177	. . . . 5 ⊢ (𝐴 ∈ 𝑦 → (∃𝑥 ∈ On (𝑅1‘𝑥) = 𝑦 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)))
16		r1fun 9795	. . . . . 6 ⊢ Fun 𝑅1
17		fvelima 6970	. . . . . 6 ⊢ ((Fun 𝑅1 ∧ 𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On (𝑅1‘𝑥) = 𝑦)
18	16, 17	mpan 703	. . . . 5 ⊢ (𝑦 ∈ (𝑅1 “ On) → ∃𝑥 ∈ On (𝑅1‘𝑥) = 𝑦)
19	15, 18	impel 515	. . . 4 ⊢ ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
20	19	exlimiv 1965	. . 3 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
21	1, 20	sylbi 220	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
22		elfvdm 6939	. . . . . 6 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
23		fvelrn 7097	. . . . . 6 ⊢ ((Fun 𝑅1 ∧ suc 𝑥 ∈ dom 𝑅1) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
24	16, 22, 23	sylancr 599	. . . . 5 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
25		rnr1 9798	. . . . 5 ⊢ ran 𝑅1 = (𝑅1 “ On)
26	24, 25	eleqtrdi 2874	. . . 4 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On))
27		elunii 4875	. . . 4 ⊢ ((𝐴 ∈ (𝑅1‘suc 𝑥) ∧ (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On)) → 𝐴 ∈ ∪ (𝑅1 “ On))
28	26, 27	mpdan 700	. . 3 ⊢ (𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ ∪ (𝑅1 “ On))
29	28	rexlimivw 3163	. 2 ⊢ (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 ∈ ∪ (𝑅1 “ On))
30	21, 29	impbii 212	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900  𝒫 cpw 4560  ∪ cuni 4870  Tr wtr 5215  dom cdm 5655  ran crn 5656   “ cima 5658  Oncon0 6377  suc csuc 6379  Fun wfun 6549  ‘cfv 6555  𝑅₁cr1 9790

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-om 7899  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-r1 9792

This theorem is used by:  rankf  9826  r1elwf  9828  rankvalb  9829  rankidb  9832  rankwflem  9848  tcrank  9925  dfac12r  10249