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Mirrors > Home > MPE Home > Th. List > Mathboxes > elhf | Structured version Visualization version GIF version |
Description: Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015.) |
Ref | Expression |
---|---|
elhf | ⊢ (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-hf 33634 | . . 3 ⊢ Hf = ∪ (𝑅1 “ ω) | |
2 | 1 | eleq2i 2904 | . 2 ⊢ (𝐴 ∈ Hf ↔ 𝐴 ∈ ∪ (𝑅1 “ ω)) |
3 | r111 9203 | . . 3 ⊢ 𝑅1:On–1-1→V | |
4 | f1fun 6576 | . . 3 ⊢ (𝑅1:On–1-1→V → Fun 𝑅1) | |
5 | eluniima 7008 | . . 3 ⊢ (Fun 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))) | |
6 | 3, 4, 5 | mp2b 10 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥)) |
7 | 2, 6 | bitri 277 | 1 ⊢ (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2110 ∃wrex 3139 Vcvv 3494 ∪ cuni 4837 “ cima 5557 Oncon0 6190 Fun wfun 6348 –1-1→wf1 6351 ‘cfv 6354 ωcom 7579 𝑅1cr1 9190 Hf chf 33633 |
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 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
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 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-r1 9192 df-hf 33634 |
This theorem is referenced by: elhf2 33636 0hf 33638 |
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