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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 34973 | . . 3 ⊢ Hf = ∪ (𝑅1 “ ω) | |
2 | 1 | eleq2i 2824 | . 2 ⊢ (𝐴 ∈ Hf ↔ 𝐴 ∈ ∪ (𝑅1 “ ω)) |
3 | r111 9752 | . . 3 ⊢ 𝑅1:On–1-1→V | |
4 | f1fun 6776 | . . 3 ⊢ (𝑅1:On–1-1→V → Fun 𝑅1) | |
5 | eluniima 7233 | . . 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 274 | 1 ⊢ (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2106 ∃wrex 3069 Vcvv 3473 ∪ cuni 4901 “ cima 5672 Oncon0 6353 Fun wfun 6526 –1-1→wf1 6529 ‘cfv 6532 ωcom 7838 𝑅1cr1 9739 Hf chf 34972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-r1 9741 df-hf 34973 |
This theorem is referenced by: elhf2 34975 0hf 34977 |
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