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Theorem fin2so 35317
 Description: Any totally ordered Tarski-finite set is finite; in particular, no amorphous set can be ordered. Theorem 2 of [Levy58]] p. 4. (Contributed by Brendan Leahy, 28-Jun-2019.)
Assertion
Ref Expression
fin2so ((𝐴 ∈ FinII𝑅 Or 𝐴) → 𝐴 ∈ Fin)

Proof of Theorem fin2so
Dummy variables 𝑣 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplll 775 . . . . . . . . . . . 12 ((((𝐴 ∈ FinII𝑅 Or 𝐴) ∧ 𝑥𝐴) ∧ 𝑥 ≠ ∅) → 𝐴 ∈ FinII)
2 ssrab2 3985 . . . . . . . . . . . . . . . . . . 19 {𝑤𝑥𝑤𝑅𝑣} ⊆ 𝑥
3 sstr 3901 . . . . . . . . . . . . . . . . . . 19 (({𝑤𝑥𝑤𝑅𝑣} ⊆ 𝑥𝑥𝐴) → {𝑤𝑥𝑤𝑅𝑣} ⊆ 𝐴)
42, 3mpan 690 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → {𝑤𝑥𝑤𝑅𝑣} ⊆ 𝐴)
5 elpw2g 5215 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ FinII → ({𝑤𝑥𝑤𝑅𝑣} ∈ 𝒫 𝐴 ↔ {𝑤𝑥𝑤𝑅𝑣} ⊆ 𝐴))
65biimpar 482 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ FinII ∧ {𝑤𝑥𝑤𝑅𝑣} ⊆ 𝐴) → {𝑤𝑥𝑤𝑅𝑣} ∈ 𝒫 𝐴)
74, 6sylan2 596 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ FinII𝑥𝐴) → {𝑤𝑥𝑤𝑅𝑣} ∈ 𝒫 𝐴)
87ralrimivw 3115 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ FinII𝑥𝐴) → ∀𝑣𝑥 {𝑤𝑥𝑤𝑅𝑣} ∈ 𝒫 𝐴)
9 vex 3414 . . . . . . . . . . . . . . . . . . 19 𝑥 ∈ V
109rabex 5203 . . . . . . . . . . . . . . . . . 18 {𝑤𝑥𝑤𝑅𝑣} ∈ V
1110rgenw 3083 . . . . . . . . . . . . . . . . 17 𝑣𝑥 {𝑤𝑥𝑤𝑅𝑣} ∈ V
12 eqid 2759 . . . . . . . . . . . . . . . . . 18 (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})
13 eleq1 2840 . . . . . . . . . . . . . . . . . 18 (𝑦 = {𝑤𝑥𝑤𝑅𝑣} → (𝑦 ∈ 𝒫 𝐴 ↔ {𝑤𝑥𝑤𝑅𝑣} ∈ 𝒫 𝐴))
1412, 13ralrnmptw 6852 . . . . . . . . . . . . . . . . 17 (∀𝑣𝑥 {𝑤𝑥𝑤𝑅𝑣} ∈ V → (∀𝑦 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})𝑦 ∈ 𝒫 𝐴 ↔ ∀𝑣𝑥 {𝑤𝑥𝑤𝑅𝑣} ∈ 𝒫 𝐴))
1511, 14ax-mp 5 . . . . . . . . . . . . . . . 16 (∀𝑦 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})𝑦 ∈ 𝒫 𝐴 ↔ ∀𝑣𝑥 {𝑤𝑥𝑤𝑅𝑣} ∈ 𝒫 𝐴)
168, 15sylibr 237 . . . . . . . . . . . . . . 15 ((𝐴 ∈ FinII𝑥𝐴) → ∀𝑦 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})𝑦 ∈ 𝒫 𝐴)
17 dfss3 3881 . . . . . . . . . . . . . . 15 (ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ⊆ 𝒫 𝐴 ↔ ∀𝑦 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})𝑦 ∈ 𝒫 𝐴)
1816, 17sylibr 237 . . . . . . . . . . . . . 14 ((𝐴 ∈ FinII𝑥𝐴) → ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ⊆ 𝒫 𝐴)
1918adantlr 715 . . . . . . . . . . . . 13 (((𝐴 ∈ FinII𝑅 Or 𝐴) ∧ 𝑥𝐴) → ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ⊆ 𝒫 𝐴)
2019adantr 485 . . . . . . . . . . . 12 ((((𝐴 ∈ FinII𝑅 Or 𝐴) ∧ 𝑥𝐴) ∧ 𝑥 ≠ ∅) → ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ⊆ 𝒫 𝐴)
2110, 12dmmpti 6476 . . . . . . . . . . . . . . 15 dom (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = 𝑥
2221neeq1i 3016 . . . . . . . . . . . . . 14 (dom (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ≠ ∅ ↔ 𝑥 ≠ ∅)
23 dm0rn0 5767 . . . . . . . . . . . . . . 15 (dom (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = ∅ ↔ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = ∅)
2423necon3bii 3004 . . . . . . . . . . . . . 14 (dom (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ≠ ∅ ↔ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ≠ ∅)
2522, 24sylbb1 240 . . . . . . . . . . . . 13 (𝑥 ≠ ∅ → ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ≠ ∅)
2625adantl 486 . . . . . . . . . . . 12 ((((𝐴 ∈ FinII𝑅 Or 𝐴) ∧ 𝑥𝐴) ∧ 𝑥 ≠ ∅) → ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ≠ ∅)
27 soss 5463 . . . . . . . . . . . . . . . 16 (𝑥𝐴 → (𝑅 Or 𝐴𝑅 Or 𝑥))
2827impcom 412 . . . . . . . . . . . . . . 15 ((𝑅 Or 𝐴𝑥𝐴) → 𝑅 Or 𝑥)
29 porpss 7452 . . . . . . . . . . . . . . . . 17 [] Po ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})
3029a1i 11 . . . . . . . . . . . . . . . 16 (𝑅 Or 𝑥 → [] Po ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}))
31 solin 5468 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 Or 𝑥 ∧ (𝑣𝑥𝑦𝑥)) → (𝑣𝑅𝑦𝑣 = 𝑦𝑦𝑅𝑣))
32 fin2solem 35316 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 Or 𝑥 ∧ (𝑣𝑥𝑦𝑥)) → (𝑣𝑅𝑦 → {𝑤𝑥𝑤𝑅𝑣} [] {𝑤𝑥𝑤𝑅𝑦}))
33 breq2 5037 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑣 = 𝑦 → (𝑤𝑅𝑣𝑤𝑅𝑦))
3433rabbidv 3393 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = 𝑦 → {𝑤𝑥𝑤𝑅𝑣} = {𝑤𝑥𝑤𝑅𝑦})
3534a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 Or 𝑥 ∧ (𝑣𝑥𝑦𝑥)) → (𝑣 = 𝑦 → {𝑤𝑥𝑤𝑅𝑣} = {𝑤𝑥𝑤𝑅𝑦}))
36 fin2solem 35316 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑅 Or 𝑥 ∧ (𝑦𝑥𝑣𝑥)) → (𝑦𝑅𝑣 → {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑣}))
3736ancom2s 650 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 Or 𝑥 ∧ (𝑣𝑥𝑦𝑥)) → (𝑦𝑅𝑣 → {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑣}))
3832, 35, 373orim123d 1442 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 Or 𝑥 ∧ (𝑣𝑥𝑦𝑥)) → ((𝑣𝑅𝑦𝑣 = 𝑦𝑦𝑅𝑣) → ({𝑤𝑥𝑤𝑅𝑣} [] {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑣} = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑣})))
3931, 38mpd 15 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 Or 𝑥 ∧ (𝑣𝑥𝑦𝑥)) → ({𝑤𝑥𝑤𝑅𝑣} [] {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑣} = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑣}))
4039ralrimivva 3121 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 Or 𝑥 → ∀𝑣𝑥𝑦𝑥 ({𝑤𝑥𝑤𝑅𝑣} [] {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑣} = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑣}))
41 breq1 5036 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑢 = {𝑤𝑥𝑤𝑅𝑣} → (𝑢 [] {𝑤𝑥𝑤𝑅𝑦} ↔ {𝑤𝑥𝑤𝑅𝑣} [] {𝑤𝑥𝑤𝑅𝑦}))
42 eqeq1 2763 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑢 = {𝑤𝑥𝑤𝑅𝑣} → (𝑢 = {𝑤𝑥𝑤𝑅𝑦} ↔ {𝑤𝑥𝑤𝑅𝑣} = {𝑤𝑥𝑤𝑅𝑦}))
43 breq2 5037 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑢 = {𝑤𝑥𝑤𝑅𝑣} → ({𝑤𝑥𝑤𝑅𝑦} [] 𝑢 ↔ {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑣}))
4441, 42, 433orbi123d 1433 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢 = {𝑤𝑥𝑤𝑅𝑣} → ((𝑢 [] {𝑤𝑥𝑤𝑅𝑦} ∨ 𝑢 = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] 𝑢) ↔ ({𝑤𝑥𝑤𝑅𝑣} [] {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑣} = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑣})))
4544ralbidv 3127 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = {𝑤𝑥𝑤𝑅𝑣} → (∀𝑦𝑥 (𝑢 [] {𝑤𝑥𝑤𝑅𝑦} ∨ 𝑢 = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] 𝑢) ↔ ∀𝑦𝑥 ({𝑤𝑥𝑤𝑅𝑣} [] {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑣} = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑣})))
4612, 45ralrnmptw 6852 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑣𝑥 {𝑤𝑥𝑤𝑅𝑣} ∈ V → (∀𝑢 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})∀𝑦𝑥 (𝑢 [] {𝑤𝑥𝑤𝑅𝑦} ∨ 𝑢 = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] 𝑢) ↔ ∀𝑣𝑥𝑦𝑥 ({𝑤𝑥𝑤𝑅𝑣} [] {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑣} = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑣})))
4711, 46ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑢 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})∀𝑦𝑥 (𝑢 [] {𝑤𝑥𝑤𝑅𝑦} ∨ 𝑢 = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] 𝑢) ↔ ∀𝑣𝑥𝑦𝑥 ({𝑤𝑥𝑤𝑅𝑣} [] {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑣} = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] {𝑤𝑥𝑤𝑅𝑣}))
4840, 47sylibr 237 . . . . . . . . . . . . . . . . . . . 20 (𝑅 Or 𝑥 → ∀𝑢 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})∀𝑦𝑥 (𝑢 [] {𝑤𝑥𝑤𝑅𝑦} ∨ 𝑢 = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] 𝑢))
4948r19.21bi 3138 . . . . . . . . . . . . . . . . . . 19 ((𝑅 Or 𝑥𝑢 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})) → ∀𝑦𝑥 (𝑢 [] {𝑤𝑥𝑤𝑅𝑦} ∨ 𝑢 = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] 𝑢))
509rabex 5203 . . . . . . . . . . . . . . . . . . . . 21 {𝑤𝑥𝑤𝑅𝑦} ∈ V
5150rgenw 3083 . . . . . . . . . . . . . . . . . . . 20 𝑦𝑥 {𝑤𝑥𝑤𝑅𝑦} ∈ V
5234cbvmptv 5136 . . . . . . . . . . . . . . . . . . . . 21 (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = (𝑦𝑥 ↦ {𝑤𝑥𝑤𝑅𝑦})
53 breq2 5037 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = {𝑤𝑥𝑤𝑅𝑦} → (𝑢 [] 𝑧𝑢 [] {𝑤𝑥𝑤𝑅𝑦}))
54 eqeq2 2771 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = {𝑤𝑥𝑤𝑅𝑦} → (𝑢 = 𝑧𝑢 = {𝑤𝑥𝑤𝑅𝑦}))
55 breq1 5036 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = {𝑤𝑥𝑤𝑅𝑦} → (𝑧 [] 𝑢 ↔ {𝑤𝑥𝑤𝑅𝑦} [] 𝑢))
5653, 54, 553orbi123d 1433 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = {𝑤𝑥𝑤𝑅𝑦} → ((𝑢 [] 𝑧𝑢 = 𝑧𝑧 [] 𝑢) ↔ (𝑢 [] {𝑤𝑥𝑤𝑅𝑦} ∨ 𝑢 = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] 𝑢)))
5752, 56ralrnmptw 6852 . . . . . . . . . . . . . . . . . . . 20 (∀𝑦𝑥 {𝑤𝑥𝑤𝑅𝑦} ∈ V → (∀𝑧 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})(𝑢 [] 𝑧𝑢 = 𝑧𝑧 [] 𝑢) ↔ ∀𝑦𝑥 (𝑢 [] {𝑤𝑥𝑤𝑅𝑦} ∨ 𝑢 = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] 𝑢)))
5851, 57ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (∀𝑧 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})(𝑢 [] 𝑧𝑢 = 𝑧𝑧 [] 𝑢) ↔ ∀𝑦𝑥 (𝑢 [] {𝑤𝑥𝑤𝑅𝑦} ∨ 𝑢 = {𝑤𝑥𝑤𝑅𝑦} ∨ {𝑤𝑥𝑤𝑅𝑦} [] 𝑢))
5949, 58sylibr 237 . . . . . . . . . . . . . . . . . 18 ((𝑅 Or 𝑥𝑢 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})) → ∀𝑧 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})(𝑢 [] 𝑧𝑢 = 𝑧𝑧 [] 𝑢))
6059r19.21bi 3138 . . . . . . . . . . . . . . . . 17 (((𝑅 Or 𝑥𝑢 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})) ∧ 𝑧 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})) → (𝑢 [] 𝑧𝑢 = 𝑧𝑧 [] 𝑢))
6160anasss 471 . . . . . . . . . . . . . . . 16 ((𝑅 Or 𝑥 ∧ (𝑢 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ∧ 𝑧 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}))) → (𝑢 [] 𝑧𝑢 = 𝑧𝑧 [] 𝑢))
6230, 61issod 5476 . . . . . . . . . . . . . . 15 (𝑅 Or 𝑥 → [] Or ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}))
6328, 62syl 17 . . . . . . . . . . . . . 14 ((𝑅 Or 𝐴𝑥𝐴) → [] Or ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}))
6463adantll 714 . . . . . . . . . . . . 13 (((𝐴 ∈ FinII𝑅 Or 𝐴) ∧ 𝑥𝐴) → [] Or ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}))
6564adantr 485 . . . . . . . . . . . 12 ((((𝐴 ∈ FinII𝑅 Or 𝐴) ∧ 𝑥𝐴) ∧ 𝑥 ≠ ∅) → [] Or ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}))
66 fin2i2 9771 . . . . . . . . . . . 12 (((𝐴 ∈ FinII ∧ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ⊆ 𝒫 𝐴) ∧ (ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ≠ ∅ ∧ [] Or ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}))) → ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}))
671, 20, 26, 65, 66syl22anc 838 . . . . . . . . . . 11 ((((𝐴 ∈ FinII𝑅 Or 𝐴) ∧ 𝑥𝐴) ∧ 𝑥 ≠ ∅) → ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}))
6852, 50elrnmpti 5802 . . . . . . . . . . 11 ( ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ↔ ∃𝑦𝑥 ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦})
6967, 68sylib 221 . . . . . . . . . 10 ((((𝐴 ∈ FinII𝑅 Or 𝐴) ∧ 𝑥𝐴) ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥 ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦})
70 ssel2 3888 . . . . . . . . . . . . . . . . . . . 20 ((𝑥𝐴𝑧𝑥) → 𝑧𝐴)
71 sonr 5466 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 Or 𝐴𝑧𝐴) → ¬ 𝑧𝑅𝑧)
7270, 71sylan2 596 . . . . . . . . . . . . . . . . . . 19 ((𝑅 Or 𝐴 ∧ (𝑥𝐴𝑧𝑥)) → ¬ 𝑧𝑅𝑧)
7372anassrs 472 . . . . . . . . . . . . . . . . . 18 (((𝑅 Or 𝐴𝑥𝐴) ∧ 𝑧𝑥) → ¬ 𝑧𝑅𝑧)
7473adantlr 715 . . . . . . . . . . . . . . . . 17 ((((𝑅 Or 𝐴𝑥𝐴) ∧ 𝑦𝑥) ∧ 𝑧𝑥) → ¬ 𝑧𝑅𝑧)
7574adantr 485 . . . . . . . . . . . . . . . 16 (((((𝑅 Or 𝐴𝑥𝐴) ∧ 𝑦𝑥) ∧ 𝑧𝑥) ∧ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦}) → ¬ 𝑧𝑅𝑧)
76 breq1 5036 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑧 → (𝑤𝑅𝑦𝑧𝑅𝑦))
7776elrab 3603 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ {𝑤𝑥𝑤𝑅𝑦} ↔ (𝑧𝑥𝑧𝑅𝑦))
7877simplbi2 505 . . . . . . . . . . . . . . . . . . 19 (𝑧𝑥 → (𝑧𝑅𝑦𝑧 ∈ {𝑤𝑥𝑤𝑅𝑦}))
7978ad2antlr 727 . . . . . . . . . . . . . . . . . 18 (((𝑦𝑥𝑧𝑥) ∧ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦}) → (𝑧𝑅𝑦𝑧 ∈ {𝑤𝑥𝑤𝑅𝑦}))
80 vex 3414 . . . . . . . . . . . . . . . . . . . . . . 23 𝑧 ∈ V
8180elint2 4846 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ↔ ∀𝑦 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})𝑧𝑦)
82 eleq2 2841 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = {𝑤𝑥𝑤𝑅𝑣} → (𝑧𝑦𝑧 ∈ {𝑤𝑥𝑤𝑅𝑣}))
8312, 82ralrnmptw 6852 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑣𝑥 {𝑤𝑥𝑤𝑅𝑣} ∈ V → (∀𝑦 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})𝑧𝑦 ↔ ∀𝑣𝑥 𝑧 ∈ {𝑤𝑥𝑤𝑅𝑣}))
8411, 83ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑦 ∈ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣})𝑧𝑦 ↔ ∀𝑣𝑥 𝑧 ∈ {𝑤𝑥𝑤𝑅𝑣})
8581, 84bitri 278 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ↔ ∀𝑣𝑥 𝑧 ∈ {𝑤𝑥𝑤𝑅𝑣})
86 breq2 5037 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑣 = 𝑧 → (𝑤𝑅𝑣𝑤𝑅𝑧))
8786rabbidv 3393 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = 𝑧 → {𝑤𝑥𝑤𝑅𝑣} = {𝑤𝑥𝑤𝑅𝑧})
8887eleq2d 2838 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = 𝑧 → (𝑧 ∈ {𝑤𝑥𝑤𝑅𝑣} ↔ 𝑧 ∈ {𝑤𝑥𝑤𝑅𝑧}))
8988rspcv 3537 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧𝑥 → (∀𝑣𝑥 𝑧 ∈ {𝑤𝑥𝑤𝑅𝑣} → 𝑧 ∈ {𝑤𝑥𝑤𝑅𝑧}))
90 breq1 5036 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 𝑧 → (𝑤𝑅𝑧𝑧𝑅𝑧))
9190elrab 3603 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ {𝑤𝑥𝑤𝑅𝑧} ↔ (𝑧𝑥𝑧𝑅𝑧))
9291simprbi 501 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ {𝑤𝑥𝑤𝑅𝑧} → 𝑧𝑅𝑧)
9389, 92syl6 35 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑥 → (∀𝑣𝑥 𝑧 ∈ {𝑤𝑥𝑤𝑅𝑣} → 𝑧𝑅𝑧))
9493adantl 486 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝑥𝑧𝑥) → (∀𝑣𝑥 𝑧 ∈ {𝑤𝑥𝑤𝑅𝑣} → 𝑧𝑅𝑧))
9585, 94syl5bi 245 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑥𝑧𝑥) → (𝑧 ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) → 𝑧𝑅𝑧))
96 eleq2 2841 . . . . . . . . . . . . . . . . . . . . 21 ( ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦} → (𝑧 ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) ↔ 𝑧 ∈ {𝑤𝑥𝑤𝑅𝑦}))
9796imbi1d 346 . . . . . . . . . . . . . . . . . . . 20 ( ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦} → ((𝑧 ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) → 𝑧𝑅𝑧) ↔ (𝑧 ∈ {𝑤𝑥𝑤𝑅𝑦} → 𝑧𝑅𝑧)))
9895, 97syl5ibcom 248 . . . . . . . . . . . . . . . . . . 19 ((𝑦𝑥𝑧𝑥) → ( ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦} → (𝑧 ∈ {𝑤𝑥𝑤𝑅𝑦} → 𝑧𝑅𝑧)))
9998imp 411 . . . . . . . . . . . . . . . . . 18 (((𝑦𝑥𝑧𝑥) ∧ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦}) → (𝑧 ∈ {𝑤𝑥𝑤𝑅𝑦} → 𝑧𝑅𝑧))
10079, 99syld 47 . . . . . . . . . . . . . . . . 17 (((𝑦𝑥𝑧𝑥) ∧ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦}) → (𝑧𝑅𝑦𝑧𝑅𝑧))
101100adantlll 718 . . . . . . . . . . . . . . . 16 (((((𝑅 Or 𝐴𝑥𝐴) ∧ 𝑦𝑥) ∧ 𝑧𝑥) ∧ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦}) → (𝑧𝑅𝑦𝑧𝑅𝑧))
10275, 101mtod 201 . . . . . . . . . . . . . . 15 (((((𝑅 Or 𝐴𝑥𝐴) ∧ 𝑦𝑥) ∧ 𝑧𝑥) ∧ ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦}) → ¬ 𝑧𝑅𝑦)
103102ex 417 . . . . . . . . . . . . . 14 ((((𝑅 Or 𝐴𝑥𝐴) ∧ 𝑦𝑥) ∧ 𝑧𝑥) → ( ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦} → ¬ 𝑧𝑅𝑦))
104103ralrimdva 3119 . . . . . . . . . . . . 13 (((𝑅 Or 𝐴𝑥𝐴) ∧ 𝑦𝑥) → ( ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦} → ∀𝑧𝑥 ¬ 𝑧𝑅𝑦))
105104reximdva 3199 . . . . . . . . . . . 12 ((𝑅 Or 𝐴𝑥𝐴) → (∃𝑦𝑥 ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦} → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
106105adantll 714 . . . . . . . . . . 11 (((𝐴 ∈ FinII𝑅 Or 𝐴) ∧ 𝑥𝐴) → (∃𝑦𝑥 ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦} → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
107106adantr 485 . . . . . . . . . 10 ((((𝐴 ∈ FinII𝑅 Or 𝐴) ∧ 𝑥𝐴) ∧ 𝑥 ≠ ∅) → (∃𝑦𝑥 ran (𝑣𝑥 ↦ {𝑤𝑥𝑤𝑅𝑣}) = {𝑤𝑥𝑤𝑅𝑦} → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
10869, 107mpd 15 . . . . . . . . 9 ((((𝐴 ∈ FinII𝑅 Or 𝐴) ∧ 𝑥𝐴) ∧ 𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)
109108expl 462 . . . . . . . 8 ((𝐴 ∈ FinII𝑅 Or 𝐴) → ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
110109alrimiv 1929 . . . . . . 7 ((𝐴 ∈ FinII𝑅 Or 𝐴) → ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
111 df-fr 5484 . . . . . . 7 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
112110, 111sylibr 237 . . . . . 6 ((𝐴 ∈ FinII𝑅 Or 𝐴) → 𝑅 Fr 𝐴)
113 simpr 489 . . . . . 6 ((𝐴 ∈ FinII𝑅 Or 𝐴) → 𝑅 Or 𝐴)
114 df-we 5486 . . . . . 6 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
115112, 113, 114sylanbrc 587 . . . . 5 ((𝐴 ∈ FinII𝑅 Or 𝐴) → 𝑅 We 𝐴)
116 weinxp 5606 . . . . 5 (𝑅 We 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) We 𝐴)
117115, 116sylib 221 . . . 4 ((𝐴 ∈ FinII𝑅 Or 𝐴) → (𝑅 ∩ (𝐴 × 𝐴)) We 𝐴)
118 sqxpexg 7477 . . . . . 6 (𝐴 ∈ FinII → (𝐴 × 𝐴) ∈ V)
119 incom 4107 . . . . . . 7 (𝑅 ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ 𝑅)
120 inex1g 5190 . . . . . . 7 ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ 𝑅) ∈ V)
121119, 120eqeltrid 2857 . . . . . 6 ((𝐴 × 𝐴) ∈ V → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
122 weeq1 5513 . . . . . . 7 (𝑧 = (𝑅 ∩ (𝐴 × 𝐴)) → (𝑧 We 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) We 𝐴))
123122spcegv 3516 . . . . . 6 ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → ((𝑅 ∩ (𝐴 × 𝐴)) We 𝐴 → ∃𝑧 𝑧 We 𝐴))
124118, 121, 1233syl 18 . . . . 5 (𝐴 ∈ FinII → ((𝑅 ∩ (𝐴 × 𝐴)) We 𝐴 → ∃𝑧 𝑧 We 𝐴))
125124imp 411 . . . 4 ((𝐴 ∈ FinII ∧ (𝑅 ∩ (𝐴 × 𝐴)) We 𝐴) → ∃𝑧 𝑧 We 𝐴)
126117, 125syldan 595 . . 3 ((𝐴 ∈ FinII𝑅 Or 𝐴) → ∃𝑧 𝑧 We 𝐴)
127 ween 9488 . . 3 (𝐴 ∈ dom card ↔ ∃𝑧 𝑧 We 𝐴)
128126, 127sylibr 237 . 2 ((𝐴 ∈ FinII𝑅 Or 𝐴) → 𝐴 ∈ dom card)
129 fin23 9842 . . . . 5 (𝐴 ∈ FinII𝐴 ∈ FinIII)
130 fin34 9843 . . . . 5 (𝐴 ∈ FinIII𝐴 ∈ FinIV)
131 fin45 9845 . . . . 5 (𝐴 ∈ FinIV𝐴 ∈ FinV)
132129, 130, 1313syl 18 . . . 4 (𝐴 ∈ FinII𝐴 ∈ FinV)
133 fin56 9846 . . . 4 (𝐴 ∈ FinV𝐴 ∈ FinVI)
134 fin67 9848 . . . 4 (𝐴 ∈ FinVI𝐴 ∈ FinVII)
135132, 133, 1343syl 18 . . 3 (𝐴 ∈ FinII𝐴 ∈ FinVII)
136 fin71num 9850 . . . 4 (𝐴 ∈ dom card → (𝐴 ∈ FinVII𝐴 ∈ Fin))
137136biimpac 483 . . 3 ((𝐴 ∈ FinVII𝐴 ∈ dom card) → 𝐴 ∈ Fin)
138135, 137sylan 584 . 2 ((𝐴 ∈ FinII𝐴 ∈ dom card) → 𝐴 ∈ Fin)
139128, 138syldan 595 1 ((𝐴 ∈ FinII𝑅 Or 𝐴) → 𝐴 ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ w3o 1084  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2112   ≠ wne 2952  ∀wral 3071  ∃wrex 3072  {crab 3075  Vcvv 3410   ∩ cin 3858   ⊆ wss 3859  ∅c0 4226  𝒫 cpw 4495  ∩ cint 4839   class class class wbr 5033   ↦ cmpt 5113   Po wpo 5442   Or wor 5443   Fr wfr 5481   We wwe 5483   × cxp 5523  dom cdm 5525  ran crn 5526   [⊊] crpss 7447  Fincfn 8528  cardccrd 9390  FinIIcfin2 9732  FinIVcfin4 9733  FinIIIcfin3 9734  FinVcfin5 9735  FinVIcfin6 9736  FinVIIcfin7 9737 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-inf2 9130 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-se 5485  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-isom 6345  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-rpss 7448  df-om 7581  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-seqom 8095  df-1o 8113  df-2o 8114  df-oadd 8117  df-er 8300  df-map 8419  df-en 8529  df-dom 8530  df-sdom 8531  df-fin 8532  df-oi 9000  df-wdom 9055  df-dju 9356  df-card 9394  df-fin2 9739  df-fin4 9740  df-fin3 9741  df-fin5 9742  df-fin6 9743  df-fin7 9744 This theorem is referenced by: (None)
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