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Theorem rankwflem 8977

Description: Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 8954 is useful in proofs of theorems about the rank function. (Contributed by NM, 4-Oct-2003.)

**Assertion**
Ref	Expression
rankwflem	⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅₁‘suc 𝑥))

Distinct variable group: 𝑥,𝐴 Allowed substitution hint: 𝑉(𝑥)

**Proof of Theorem rankwflem**
Step	Hyp	Ref	Expression
1		elex 3414	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		unir1 8975	. . 3 ⊢ ∪ (𝑅₁ “ On) = V
3	1, 2	syl6eleqr 2870	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ∪ (𝑅₁ “ On))
4		rankwflemb 8955	. 2 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅₁‘suc 𝑥))
5	3, 4	sylib 210	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅₁‘suc 𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2107 ∃wrex 3091 Vcvv 3398 ∪ cuni 4673 “ cima 5360 Oncon0 5978 suc csuc 5980 ‘cfv 6137 𝑅₁cr1 8924

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-reg 8788 ax-inf2 8837

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-r1 8926

This theorem is referenced by: (None)

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