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Mirrors > Home > MPE Home > Th. List > setind | Structured version Visualization version GIF version |
Description: Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.) |
Ref | Expression |
---|---|
setind | ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssindif0 4364 | . . . . . . 7 ⊢ (𝑦 ⊆ 𝐴 ↔ (𝑦 ∩ (V ∖ 𝐴)) = ∅) | |
2 | sseq1 3912 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴)) | |
3 | eleq1w 2813 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
4 | 2, 3 | imbi12d 348 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ↔ (𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝐴))) |
5 | 4 | spvv 2006 | . . . . . . 7 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝐴)) |
6 | 1, 5 | syl5bir 246 | . . . . . 6 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → ((𝑦 ∩ (V ∖ 𝐴)) = ∅ → 𝑦 ∈ 𝐴)) |
7 | eldifn 4028 | . . . . . 6 ⊢ (𝑦 ∈ (V ∖ 𝐴) → ¬ 𝑦 ∈ 𝐴) | |
8 | 6, 7 | nsyli 160 | . . . . 5 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (𝑦 ∈ (V ∖ 𝐴) → ¬ (𝑦 ∩ (V ∖ 𝐴)) = ∅)) |
9 | 8 | imp 410 | . . . 4 ⊢ ((∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (V ∖ 𝐴)) → ¬ (𝑦 ∩ (V ∖ 𝐴)) = ∅) |
10 | 9 | nrexdv 3179 | . . 3 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → ¬ ∃𝑦 ∈ (V ∖ 𝐴)(𝑦 ∩ (V ∖ 𝐴)) = ∅) |
11 | zfregs 9326 | . . . 4 ⊢ ((V ∖ 𝐴) ≠ ∅ → ∃𝑦 ∈ (V ∖ 𝐴)(𝑦 ∩ (V ∖ 𝐴)) = ∅) | |
12 | 11 | necon1bi 2960 | . . 3 ⊢ (¬ ∃𝑦 ∈ (V ∖ 𝐴)(𝑦 ∩ (V ∖ 𝐴)) = ∅ → (V ∖ 𝐴) = ∅) |
13 | 10, 12 | syl 17 | . 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (V ∖ 𝐴) = ∅) |
14 | vdif0 4369 | . 2 ⊢ (𝐴 = V ↔ (V ∖ 𝐴) = ∅) | |
15 | 13, 14 | sylibr 237 | 1 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1541 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 Vcvv 3398 ∖ cdif 3850 ∩ cin 3852 ⊆ wss 3853 ∅c0 4223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 ax-reg 9186 ax-inf2 9234 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 |
This theorem is referenced by: setind2 9329 tz9.13 9372 unir1 9394 setinds 33424 vsetrec 46022 |
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