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Theorem rankwflem 9238

Description: Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 9215 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 3512	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		unir1 9236	. . 3 ⊢ ∪ (𝑅₁ “ On) = V
3	1, 2	eleqtrrdi 2924	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ∪ (𝑅₁ “ On))
4		rankwflemb 9216	. 2 ⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅₁‘suc 𝑥))
5	3, 4	sylib 220	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅₁‘suc 𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2110 ∃wrex 3139 Vcvv 3494 ∪ cuni 4831 “ cima 5552 Oncon0 6185 suc csuc 6187 ‘cfv 6349 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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-r1 9187

This theorem is referenced by: (None)

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