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Theorem dfon2lem3 30934
 Description: Lemma for dfon2 30941. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.)
Assertion
Ref Expression
dfon2lem3 (𝐴𝑉 → (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → (Tr 𝐴 ∧ ∀𝑧𝐴 ¬ 𝑧𝑧)))
Distinct variable group:   𝑥,𝐴,𝑧
Allowed substitution hints:   𝑉(𝑥,𝑧)

Proof of Theorem dfon2lem3
Dummy variables 𝑤 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 untelirr 30839 . . . . 5 (∀𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑧𝑧 → ¬ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)})
2 eluni2 4376 . . . . . 6 (𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ↔ ∃𝑥 ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}𝑧𝑥)
3 vex 3176 . . . . . . . . . 10 𝑥 ∈ V
4 sseq1 3589 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
5 treq 4686 . . . . . . . . . . 11 (𝑤 = 𝑥 → (Tr 𝑤 ↔ Tr 𝑥))
6 raleq 3115 . . . . . . . . . . 11 (𝑤 = 𝑥 → (∀𝑡𝑤 ¬ 𝑡𝑡 ↔ ∀𝑡𝑥 ¬ 𝑡𝑡))
74, 5, 63anbi123d 1391 . . . . . . . . . 10 (𝑤 = 𝑥 → ((𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡) ↔ (𝑥𝐴 ∧ Tr 𝑥 ∧ ∀𝑡𝑥 ¬ 𝑡𝑡)))
83, 7elab 3319 . . . . . . . . 9 (𝑥 ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ↔ (𝑥𝐴 ∧ Tr 𝑥 ∧ ∀𝑡𝑥 ¬ 𝑡𝑡))
9 elequ1 1984 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → (𝑡𝑡𝑧𝑡))
10 elequ2 1991 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → (𝑧𝑡𝑧𝑧))
119, 10bitrd 267 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (𝑡𝑡𝑧𝑧))
1211notbid 307 . . . . . . . . . . . 12 (𝑡 = 𝑧 → (¬ 𝑡𝑡 ↔ ¬ 𝑧𝑧))
1312cbvralv 3147 . . . . . . . . . . 11 (∀𝑡𝑥 ¬ 𝑡𝑡 ↔ ∀𝑧𝑥 ¬ 𝑧𝑧)
1413biimpi 205 . . . . . . . . . 10 (∀𝑡𝑥 ¬ 𝑡𝑡 → ∀𝑧𝑥 ¬ 𝑧𝑧)
15143ad2ant3 1077 . . . . . . . . 9 ((𝑥𝐴 ∧ Tr 𝑥 ∧ ∀𝑡𝑥 ¬ 𝑡𝑡) → ∀𝑧𝑥 ¬ 𝑧𝑧)
168, 15sylbi 206 . . . . . . . 8 (𝑥 ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → ∀𝑧𝑥 ¬ 𝑧𝑧)
17 rsp 2913 . . . . . . . 8 (∀𝑧𝑥 ¬ 𝑧𝑧 → (𝑧𝑥 → ¬ 𝑧𝑧))
1816, 17syl 17 . . . . . . 7 (𝑥 ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → (𝑧𝑥 → ¬ 𝑧𝑧))
1918rexlimiv 3009 . . . . . 6 (∃𝑥 ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}𝑧𝑥 → ¬ 𝑧𝑧)
202, 19sylbi 206 . . . . 5 (𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → ¬ 𝑧𝑧)
211, 20mprg 2910 . . . 4 ¬ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}
22 dfon2lem2 30933 . . . . 5 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴
23 dfpss2 3654 . . . . . 6 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊊ 𝐴 ↔ ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴 ∧ ¬ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = 𝐴))
24 dfon2lem1 30932 . . . . . . 7 Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}
25 ssexg 4732 . . . . . . . . . 10 (( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴𝐴𝑉) → {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ V)
2622, 25mpan 702 . . . . . . . . 9 (𝐴𝑉 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ V)
27 psseq1 3656 . . . . . . . . . . . . 13 (𝑥 = {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → (𝑥𝐴 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊊ 𝐴))
28 treq 4686 . . . . . . . . . . . . 13 (𝑥 = {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → (Tr 𝑥 ↔ Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
2927, 28anbi12d 743 . . . . . . . . . . . 12 (𝑥 = {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → ((𝑥𝐴 ∧ Tr 𝑥) ↔ ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊊ 𝐴 ∧ Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)})))
30 eleq1 2676 . . . . . . . . . . . 12 (𝑥 = {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → (𝑥𝐴 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴))
3129, 30imbi12d 333 . . . . . . . . . . 11 (𝑥 = {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → (((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ↔ (( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊊ 𝐴 ∧ Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}) → {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴)))
3231spcgv 3266 . . . . . . . . . 10 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ V → (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → (( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊊ 𝐴 ∧ Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}) → {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴)))
3332imp 444 . . . . . . . . 9 (( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ V ∧ ∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴)) → (( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊊ 𝐴 ∧ Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}) → {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴))
3426, 33sylan 487 . . . . . . . 8 ((𝐴𝑉 ∧ ∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴)) → (( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊊ 𝐴 ∧ Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}) → {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴))
35 snssi 4280 . . . . . . . . . 10 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴 → { {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}} ⊆ 𝐴)
36 unss 3749 . . . . . . . . . . 11 (( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴 ∧ { {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}} ⊆ 𝐴) ↔ ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∪ { {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}}) ⊆ 𝐴)
37 df-suc 5646 . . . . . . . . . . . 12 suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∪ { {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}})
3837sseq1i 3592 . . . . . . . . . . 11 (suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴 ↔ ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∪ { {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}}) ⊆ 𝐴)
3936, 38sylbb2 227 . . . . . . . . . 10 (( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴 ∧ { {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}} ⊆ 𝐴) → suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴)
4022, 35, 39sylancr 694 . . . . . . . . 9 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴 → suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴)
41 suctr 5725 . . . . . . . . . . . . 13 (Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → Tr suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)})
4224, 41ax-mp 5 . . . . . . . . . . . 12 Tr suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}
43 untuni 30840 . . . . . . . . . . . . . 14 (∀𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑧𝑧 ↔ ∀𝑥 ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}∀𝑧𝑥 ¬ 𝑧𝑧)
4443, 16mprgbir 2911 . . . . . . . . . . . . 13 𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑧𝑧
45 nfv 1830 . . . . . . . . . . . . . . . . 17 𝑡 𝑤𝐴
46 nfv 1830 . . . . . . . . . . . . . . . . 17 𝑡Tr 𝑤
47 nfra1 2925 . . . . . . . . . . . . . . . . 17 𝑡𝑡𝑤 ¬ 𝑡𝑡
4845, 46, 47nf3an 1819 . . . . . . . . . . . . . . . 16 𝑡(𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)
4948nfab 2755 . . . . . . . . . . . . . . 15 𝑡{𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}
5049nfuni 4378 . . . . . . . . . . . . . 14 𝑡 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}
5150untsucf 30841 . . . . . . . . . . . . 13 (∀𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑧𝑧 → ∀𝑡 ∈ suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑡𝑡)
5244, 51ax-mp 5 . . . . . . . . . . . 12 𝑡 ∈ suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑡𝑡
53 sucexg 6902 . . . . . . . . . . . . . 14 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴 → suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ V)
54 sseq1 3589 . . . . . . . . . . . . . . . . 17 (𝑧 = suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → (𝑧𝐴 ↔ suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴))
55 treq 4686 . . . . . . . . . . . . . . . . 17 (𝑧 = suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → (Tr 𝑧 ↔ Tr suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
56 nfcv 2751 . . . . . . . . . . . . . . . . . 18 𝑡𝑧
5750nfsuc 5713 . . . . . . . . . . . . . . . . . 18 𝑡 suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}
5856, 57raleqf 3111 . . . . . . . . . . . . . . . . 17 (𝑧 = suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → (∀𝑡𝑧 ¬ 𝑡𝑡 ↔ ∀𝑡 ∈ suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑡𝑡))
5954, 55, 583anbi123d 1391 . . . . . . . . . . . . . . . 16 (𝑧 = suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → ((𝑧𝐴 ∧ Tr 𝑧 ∧ ∀𝑡𝑧 ¬ 𝑡𝑡) ↔ (suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴 ∧ Tr suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∧ ∀𝑡 ∈ suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑡𝑡)))
60 sseq1 3589 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑤𝐴𝑧𝐴))
61 treq 4686 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (Tr 𝑤 ↔ Tr 𝑧))
62 raleq 3115 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (∀𝑡𝑤 ¬ 𝑡𝑡 ↔ ∀𝑡𝑧 ¬ 𝑡𝑡))
6360, 61, 623anbi123d 1391 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → ((𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡) ↔ (𝑧𝐴 ∧ Tr 𝑧 ∧ ∀𝑡𝑧 ¬ 𝑡𝑡)))
6463cbvabv 2734 . . . . . . . . . . . . . . . 16 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = {𝑧 ∣ (𝑧𝐴 ∧ Tr 𝑧 ∧ ∀𝑡𝑧 ¬ 𝑡𝑡)}
6559, 64elab2g 3322 . . . . . . . . . . . . . . 15 (suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ V → (suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ↔ (suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴 ∧ Tr suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∧ ∀𝑡 ∈ suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑡𝑡)))
6665biimprd 237 . . . . . . . . . . . . . 14 (suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ V → ((suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴 ∧ Tr suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∧ ∀𝑡 ∈ suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑡𝑡) → suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
6753, 66syl 17 . . . . . . . . . . . . 13 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴 → ((suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴 ∧ Tr suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∧ ∀𝑡 ∈ suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑡𝑡) → suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
6867com12 32 . . . . . . . . . . . 12 ((suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴 ∧ Tr suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∧ ∀𝑡 ∈ suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑡𝑡) → ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴 → suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
6942, 52, 68mp3an23 1408 . . . . . . . . . . 11 (suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴 → ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴 → suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
7069com12 32 . . . . . . . . . 10 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴 → (suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴 → suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
71 elssuni 4403 . . . . . . . . . . 11 (suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)})
72 sucssel 5736 . . . . . . . . . . 11 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴 → (suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
7371, 72syl5 33 . . . . . . . . . 10 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴 → (suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
7470, 73syld 46 . . . . . . . . 9 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴 → (suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
7540, 74mpd 15 . . . . . . . 8 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)})
7634, 75syl6 34 . . . . . . 7 ((𝐴𝑉 ∧ ∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴)) → (( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊊ 𝐴 ∧ Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}) → {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
7724, 76mpan2i 709 . . . . . 6 ((𝐴𝑉 ∧ ∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴)) → ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊊ 𝐴 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
7823, 77syl5bir 232 . . . . 5 ((𝐴𝑉 ∧ ∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴)) → (( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴 ∧ ¬ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = 𝐴) → {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
7922, 78mpani 708 . . . 4 ((𝐴𝑉 ∧ ∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴)) → (¬ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = 𝐴 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
8021, 79mt3i 140 . . 3 ((𝐴𝑉 ∧ ∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴)) → {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = 𝐴)
8124, 44pm3.2i 470 . . . 4 (Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∧ ∀𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑧𝑧)
82 treq 4686 . . . . 5 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = 𝐴 → (Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ↔ Tr 𝐴))
83 raleq 3115 . . . . 5 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = 𝐴 → (∀𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑧𝑧 ↔ ∀𝑧𝐴 ¬ 𝑧𝑧))
8482, 83anbi12d 743 . . . 4 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = 𝐴 → ((Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∧ ∀𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑧𝑧) ↔ (Tr 𝐴 ∧ ∀𝑧𝐴 ¬ 𝑧𝑧)))
8581, 84mpbii 222 . . 3 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = 𝐴 → (Tr 𝐴 ∧ ∀𝑧𝐴 ¬ 𝑧𝑧))
8680, 85syl 17 . 2 ((𝐴𝑉 ∧ ∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴)) → (Tr 𝐴 ∧ ∀𝑧𝐴 ¬ 𝑧𝑧))
8786ex 449 1 (𝐴𝑉 → (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → (Tr 𝐴 ∧ ∀𝑧𝐴 ¬ 𝑧𝑧)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540   ⊊ wpss 3541  {csn 4125  ∪ cuni 4372  Tr wtr 4680  suc csuc 5642 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-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373  df-iun 4457  df-tr 4681  df-suc 5646 This theorem is referenced by:  dfon2lem4  30935  dfon2lem5  30936  dfon2lem7  30938  dfon2lem8  30939  dfon2lem9  30940  dfon2  30941
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