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Mirrors > Home > MPE Home > Th. List > tz9.13 | Structured version Visualization version GIF version |
Description: Every set is well-founded, assuming the Axiom of Regularity. In other words, every set belongs to a layer of the cumulative hierarchy of sets. Proposition 9.13 of [TakeutiZaring] p. 78. (Contributed by NM, 23-Sep-2003.) |
Ref | Expression |
---|---|
tz9.13.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
tz9.13 | ⊢ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tz9.13.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | setind 8493 | . . . 4 ⊢ (∀𝑧(𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)}) → {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} = V) | |
3 | ssel 3562 | . . . . . . . 8 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → (𝑤 ∈ 𝑧 → 𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)})) | |
4 | vex 3176 | . . . . . . . . 9 ⊢ 𝑤 ∈ V | |
5 | eleq1 2676 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝑅1‘𝑥) ↔ 𝑤 ∈ (𝑅1‘𝑥))) | |
6 | 5 | rexbidv 3034 | . . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥))) |
7 | 4, 6 | elab 3319 | . . . . . . . 8 ⊢ (𝑤 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} ↔ ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥)) |
8 | 3, 7 | syl6ib 240 | . . . . . . 7 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → (𝑤 ∈ 𝑧 → ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥))) |
9 | 8 | ralrimiv 2948 | . . . . . 6 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → ∀𝑤 ∈ 𝑧 ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥)) |
10 | vex 3176 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
11 | 10 | tz9.12 8536 | . . . . . 6 ⊢ (∀𝑤 ∈ 𝑧 ∃𝑥 ∈ On 𝑤 ∈ (𝑅1‘𝑥) → ∃𝑥 ∈ On 𝑧 ∈ (𝑅1‘𝑥)) |
12 | 9, 11 | syl 17 | . . . . 5 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → ∃𝑥 ∈ On 𝑧 ∈ (𝑅1‘𝑥)) |
13 | eleq1 2676 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (𝑅1‘𝑥) ↔ 𝑧 ∈ (𝑅1‘𝑥))) | |
14 | 13 | rexbidv 3034 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1‘𝑥))) |
15 | 10, 14 | elab 3319 | . . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} ↔ ∃𝑥 ∈ On 𝑧 ∈ (𝑅1‘𝑥)) |
16 | 12, 15 | sylibr 223 | . . . 4 ⊢ (𝑧 ⊆ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} → 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)}) |
17 | 2, 16 | mpg 1715 | . . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} = V |
18 | 1, 17 | eleqtrri 2687 | . 2 ⊢ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} |
19 | eleq1 2676 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝑅1‘𝑥) ↔ 𝐴 ∈ (𝑅1‘𝑥))) | |
20 | 19 | rexbidv 3034 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥))) |
21 | 1, 20 | elab 3319 | . 2 ⊢ (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)} ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)) |
22 | 18, 21 | mpbi 219 | 1 ⊢ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {cab 2596 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 Oncon0 5640 ‘cfv 5804 𝑅1cr1 8508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-reg 8380 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-r1 8510 |
This theorem is referenced by: tz9.13g 8538 elhf2 31452 |
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