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Theorem dfon2lem3 32031
Description: Lemma for dfon2 32038. 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 31928 . . . . 5 (∀𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑧𝑧 → ¬ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)})
2 eluni2 4645 . . . . . 6 (𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ↔ ∃𝑥 ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}𝑧𝑥)
3 vex 3405 . . . . . . . . . 10 𝑥 ∈ V
4 sseq1 3834 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
5 treq 4963 . . . . . . . . . . 11 (𝑤 = 𝑥 → (Tr 𝑤 ↔ Tr 𝑥))
6 raleq 3338 . . . . . . . . . . 11 (𝑤 = 𝑥 → (∀𝑡𝑤 ¬ 𝑡𝑡 ↔ ∀𝑡𝑥 ¬ 𝑡𝑡))
74, 5, 63anbi123d 1553 . . . . . . . . . 10 (𝑤 = 𝑥 → ((𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡) ↔ (𝑥𝐴 ∧ Tr 𝑥 ∧ ∀𝑡𝑥 ¬ 𝑡𝑡)))
83, 7elab 3556 . . . . . . . . 9 (𝑥 ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ↔ (𝑥𝐴 ∧ Tr 𝑥 ∧ ∀𝑡𝑥 ¬ 𝑡𝑡))
9 elequ1 2164 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → (𝑡𝑡𝑧𝑡))
10 elequ2 2171 . . . . . . . . . . . . . 14 (𝑡 = 𝑧 → (𝑧𝑡𝑧𝑧))
119, 10bitrd 270 . . . . . . . . . . . . 13 (𝑡 = 𝑧 → (𝑡𝑡𝑧𝑧))
1211notbid 309 . . . . . . . . . . . 12 (𝑡 = 𝑧 → (¬ 𝑡𝑡 ↔ ¬ 𝑧𝑧))
1312cbvralv 3371 . . . . . . . . . . 11 (∀𝑡𝑥 ¬ 𝑡𝑡 ↔ ∀𝑧𝑥 ¬ 𝑧𝑧)
1413biimpi 207 . . . . . . . . . 10 (∀𝑡𝑥 ¬ 𝑡𝑡 → ∀𝑧𝑥 ¬ 𝑧𝑧)
15143ad2ant3 1158 . . . . . . . . 9 ((𝑥𝐴 ∧ Tr 𝑥 ∧ ∀𝑡𝑥 ¬ 𝑡𝑡) → ∀𝑧𝑥 ¬ 𝑧𝑧)
168, 15sylbi 208 . . . . . . . 8 (𝑥 ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → ∀𝑧𝑥 ¬ 𝑧𝑧)
17 rsp 3128 . . . . . . . 8 (∀𝑧𝑥 ¬ 𝑧𝑧 → (𝑧𝑥 → ¬ 𝑧𝑧))
1816, 17syl 17 . . . . . . 7 (𝑥 ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → (𝑧𝑥 → ¬ 𝑧𝑧))
1918rexlimiv 3226 . . . . . 6 (∃𝑥 ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}𝑧𝑥 → ¬ 𝑧𝑧)
202, 19sylbi 208 . . . . 5 (𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → ¬ 𝑧𝑧)
211, 20mprg 3125 . . . 4 ¬ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}
22 dfon2lem2 32030 . . . . 5 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴
23 dfpss2 3901 . . . . . 6 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊊ 𝐴 ↔ ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴 ∧ ¬ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = 𝐴))
24 dfon2lem1 32029 . . . . . . 7 Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}
25 ssexg 5012 . . . . . . . . . 10 (( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴𝐴𝑉) → {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ V)
2622, 25mpan 673 . . . . . . . . 9 (𝐴𝑉 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ V)
27 psseq1 3903 . . . . . . . . . . . . 13 (𝑥 = {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → (𝑥𝐴 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊊ 𝐴))
28 treq 4963 . . . . . . . . . . . . 13 (𝑥 = {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → (Tr 𝑥 ↔ Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
2927, 28anbi12d 618 . . . . . . . . . . . 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 2884 . . . . . . . . . . . 12 (𝑥 = {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → (𝑥𝐴 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴))
3129, 30imbi12d 335 . . . . . . . . . . 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 3497 . . . . . . . . . 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 395 . . . . . . . . 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 571 . . . . . . . 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 4540 . . . . . . . . . 10 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴 → { {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}} ⊆ 𝐴)
36 unss 3997 . . . . . . . . . . 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 5955 . . . . . . . . . . . 12 suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∪ { {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}})
3837sseq1i 3837 . . . . . . . . . . 11 (suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴 ↔ ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∪ { {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}}) ⊆ 𝐴)
3936, 38sylbb2 229 . . . . . . . . . 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 577 . . . . . . . . 9 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴 → suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴)
41 suctr 6033 . . . . . . . . . . . . 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 31929 . . . . . . . . . . . . . 14 (∀𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑧𝑧 ↔ ∀𝑥 ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}∀𝑧𝑥 ¬ 𝑧𝑧)
4443, 16mprgbir 3126 . . . . . . . . . . . . 13 𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑧𝑧
45 nfv 2005 . . . . . . . . . . . . . . . . 17 𝑡 𝑤𝐴
46 nfv 2005 . . . . . . . . . . . . . . . . 17 𝑡Tr 𝑤
47 nfra1 3140 . . . . . . . . . . . . . . . . 17 𝑡𝑡𝑤 ¬ 𝑡𝑡
4845, 46, 47nf3an 1993 . . . . . . . . . . . . . . . 16 𝑡(𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)
4948nfab 2964 . . . . . . . . . . . . . . 15 𝑡{𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}
5049nfuni 4647 . . . . . . . . . . . . . 14 𝑡 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}
5150untsucf 31930 . . . . . . . . . . . . 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 sseq1 3834 . . . . . . . . . . . . . . . 16 (𝑧 = suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → (𝑧𝐴 ↔ suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ⊆ 𝐴))
54 treq 4963 . . . . . . . . . . . . . . . 16 (𝑧 = suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → (Tr 𝑧 ↔ Tr suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}))
55 nfcv 2959 . . . . . . . . . . . . . . . . 17 𝑡𝑧
5650nfsuc 6021 . . . . . . . . . . . . . . . . 17 𝑡 suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)}
5755, 56raleqf 3334 . . . . . . . . . . . . . . . 16 (𝑧 = suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} → (∀𝑡𝑧 ¬ 𝑡𝑡 ↔ ∀𝑡 ∈ suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑡𝑡))
5853, 54, 573anbi123d 1553 . . . . . . . . . . . . . . 15 (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)))
59 sseq1 3834 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → (𝑤𝐴𝑧𝐴))
60 treq 4963 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → (Tr 𝑤 ↔ Tr 𝑧))
61 raleq 3338 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → (∀𝑡𝑤 ¬ 𝑡𝑡 ↔ ∀𝑡𝑧 ¬ 𝑡𝑡))
6259, 60, 613anbi123d 1553 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑧 → ((𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡) ↔ (𝑧𝐴 ∧ Tr 𝑧 ∧ ∀𝑡𝑧 ¬ 𝑡𝑡)))
6362cbvabv 2942 . . . . . . . . . . . . . . 15 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = {𝑧 ∣ (𝑧𝐴 ∧ Tr 𝑧 ∧ ∀𝑡𝑧 ¬ 𝑡𝑡)}
6458, 63elab2g 3559 . . . . . . . . . . . . . 14 (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)))
6564biimprd 239 . . . . . . . . . . . . 13 (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)}))
66 sucexg 7249 . . . . . . . . . . . . 13 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴 → suc {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ V)
6765, 66syl11 33 . . . . . . . . . . . 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)}))
6842, 52, 67mp3an23 1570 . . . . . . . . . . 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)}))
6968com12 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)}))
70 elssuni 4672 . . . . . . . . . . 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)})
71 sucssel 6042 . . . . . . . . . . 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)}))
7270, 71syl5 34 . . . . . . . . . 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)}))
7369, 72syld 47 . . . . . . . . 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)}))
7440, 73mpd 15 . . . . . . . 8 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ 𝐴 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∈ {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)})
7534, 74syl6 35 . . . . . . 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)}))
7624, 75mpan2i 680 . . . . . 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)}))
7723, 76syl5bir 234 . . . . 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)}))
7822, 77mpani 679 . . . 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)}))
7921, 78mt3i 143 . . 3 ((𝐴𝑉 ∧ ∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴)) → {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = 𝐴)
8024, 44pm3.2i 458 . . . 4 (Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ∧ ∀𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑧𝑧)
81 treq 4963 . . . . 5 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = 𝐴 → (Tr {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ↔ Tr 𝐴))
82 raleq 3338 . . . . 5 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = 𝐴 → (∀𝑧 {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} ¬ 𝑧𝑧 ↔ ∀𝑧𝐴 ¬ 𝑧𝑧))
8381, 82anbi12d 618 . . . 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)))
8480, 83mpbii 224 . . 3 ( {𝑤 ∣ (𝑤𝐴 ∧ Tr 𝑤 ∧ ∀𝑡𝑤 ¬ 𝑡𝑡)} = 𝐴 → (Tr 𝐴 ∧ ∀𝑧𝐴 ¬ 𝑧𝑧))
8579, 84syl 17 . 2 ((𝐴𝑉 ∧ ∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴)) → (Tr 𝐴 ∧ ∀𝑧𝐴 ¬ 𝑧𝑧))
8685ex 399 1 (𝐴𝑉 → (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → (Tr 𝐴 ∧ ∀𝑧𝐴 ¬ 𝑧𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1100  wal 1635   = wceq 1637  wcel 2157  {cab 2803  wral 3107  wrex 3108  Vcvv 3402  cun 3778  wss 3780  wpss 3781  {csn 4381   cuni 4641  Tr wtr 4957  suc csuc 5951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pr 5109  ax-un 7188
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-v 3404  df-sbc 3645  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-pw 4364  df-sn 4382  df-pr 4384  df-uni 4642  df-iun 4725  df-tr 4958  df-suc 5955
This theorem is referenced by:  dfon2lem4  32032  dfon2lem5  32033  dfon2lem7  32035  dfon2lem8  32036  dfon2lem9  32037  dfon2  32038
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