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Theorem minregex 42740
Description: Given any cardinal number 𝐴, there exists an argument π‘₯, which yields the least regular uncountable value of β„΅ which is greater to or equal to 𝐴. This proof uses AC. (Contributed by RP, 23-Nov-2023.)
Assertion
Ref Expression
minregex (𝐴 ∈ (ran card βˆ– Ο‰) β†’ βˆƒπ‘₯ ∈ On π‘₯ = ∩ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))})
Distinct variable group:   π‘₯,𝐴,𝑦
Proof of Theorem minregex
StepHypRef Expression
1 eldif 3950 . . . . . . 7 (𝐴 ∈ (ran card βˆ– Ο‰) ↔ (𝐴 ∈ ran card ∧ Β¬ 𝐴 ∈ Ο‰))
2 omelon 9636 . . . . . . . . . 10 Ο‰ ∈ On
3 cardon 9934 . . . . . . . . . . 11 (cardβ€˜π΄) ∈ On
4 eleq1 2813 . . . . . . . . . . 11 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) ∈ On ↔ 𝐴 ∈ On))
53, 4mpbii 232 . . . . . . . . . 10 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
6 ontri1 6388 . . . . . . . . . 10 ((Ο‰ ∈ On ∧ 𝐴 ∈ On) β†’ (Ο‰ βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ Ο‰))
72, 5, 6sylancr 586 . . . . . . . . 9 ((cardβ€˜π΄) = 𝐴 β†’ (Ο‰ βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ Ο‰))
87pm5.32i 574 . . . . . . . 8 (((cardβ€˜π΄) = 𝐴 ∧ Ο‰ βŠ† 𝐴) ↔ ((cardβ€˜π΄) = 𝐴 ∧ Β¬ 𝐴 ∈ Ο‰))
9 iscard4 42739 . . . . . . . . 9 ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 ∈ ran card)
109anbi1i 623 . . . . . . . 8 (((cardβ€˜π΄) = 𝐴 ∧ Β¬ 𝐴 ∈ Ο‰) ↔ (𝐴 ∈ ran card ∧ Β¬ 𝐴 ∈ Ο‰))
118, 10bitr2i 276 . . . . . . 7 ((𝐴 ∈ ran card ∧ Β¬ 𝐴 ∈ Ο‰) ↔ ((cardβ€˜π΄) = 𝐴 ∧ Ο‰ βŠ† 𝐴))
12 ancom 460 . . . . . . 7 (((cardβ€˜π΄) = 𝐴 ∧ Ο‰ βŠ† 𝐴) ↔ (Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴))
131, 11, 123bitri 297 . . . . . 6 (𝐴 ∈ (ran card βˆ– Ο‰) ↔ (Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴))
1413biimpi 215 . . . . 5 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ (Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴))
15 cardalephex 10080 . . . . . . . 8 (Ο‰ βŠ† 𝐴 β†’ ((cardβ€˜π΄) = 𝐴 ↔ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯)))
1615biimpa 476 . . . . . . 7 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯))
17 eqimss 4032 . . . . . . . . 9 (𝐴 = (β„΅β€˜π‘₯) β†’ 𝐴 βŠ† (β„΅β€˜π‘₯))
1817a1i 11 . . . . . . . 8 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ (𝐴 = (β„΅β€˜π‘₯) β†’ 𝐴 βŠ† (β„΅β€˜π‘₯)))
1918reximdv 3162 . . . . . . 7 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ (βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯) β†’ βˆƒπ‘₯ ∈ On 𝐴 βŠ† (β„΅β€˜π‘₯)))
2016, 19mpd 15 . . . . . 6 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ βˆƒπ‘₯ ∈ On 𝐴 βŠ† (β„΅β€˜π‘₯))
21 onintrab2 7778 . . . . . 6 (βˆƒπ‘₯ ∈ On 𝐴 βŠ† (β„΅β€˜π‘₯) ↔ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On)
2220, 21sylib 217 . . . . 5 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On)
23 simpr 484 . . . . . . 7 (((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On) β†’ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On)
24 onsuc 7792 . . . . . . 7 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On)
2523, 24syl 17 . . . . . 6 (((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On) β†’ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On)
26 eloni 6364 . . . . . . . . 9 (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ Ord ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})
2723, 26syl 17 . . . . . . . 8 (((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On) β†’ Ord ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})
28 0elsuc 7816 . . . . . . . 8 (Ord ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} β†’ βˆ… ∈ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})
2927, 28syl 17 . . . . . . 7 (((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On) β†’ βˆ… ∈ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})
30 cardaleph 10079 . . . . . . . . 9 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ 𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
3130adantr 480 . . . . . . . 8 (((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On) β†’ 𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
32 sssucid 6434 . . . . . . . . 9 ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} βŠ† suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}
33 alephord3 10068 . . . . . . . . . 10 ((∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On ∧ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On) β†’ (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} βŠ† suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ↔ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) βŠ† (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
3423, 24, 33syl2anc2 584 . . . . . . . . 9 (((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On) β†’ (∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} βŠ† suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ↔ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) βŠ† (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
3532, 34mpbii 232 . . . . . . . 8 (((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On) β†’ (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) βŠ† (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
3631, 35eqsstrd 4012 . . . . . . 7 (((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On) β†’ 𝐴 βŠ† (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
37 alephreg 10572 . . . . . . . 8 (cfβ€˜(β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})) = (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})
3837a1i 11 . . . . . . 7 (((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On) β†’ (cfβ€˜(β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})) = (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
3929, 36, 383jca 1125 . . . . . 6 (((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On) β†’ (βˆ… ∈ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∧ 𝐴 βŠ† (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∧ (cfβ€˜(β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})) = (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
4025, 39jca 511 . . . . 5 (((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) ∧ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On) β†’ (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On ∧ (βˆ… ∈ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∧ 𝐴 βŠ† (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∧ (cfβ€˜(β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})) = (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))))
4114, 22, 40syl2anc2 584 . . . 4 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On ∧ (βˆ… ∈ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∧ 𝐴 βŠ† (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∧ (cfβ€˜(β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})) = (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))))
4214, 16syl 17 . . . . . . . 8 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯))
4317a1i 11 . . . . . . . . 9 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ (𝐴 = (β„΅β€˜π‘₯) β†’ 𝐴 βŠ† (β„΅β€˜π‘₯)))
4443reximdv 3162 . . . . . . . 8 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ (βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯) β†’ βˆƒπ‘₯ ∈ On 𝐴 βŠ† (β„΅β€˜π‘₯)))
4542, 44mpd 15 . . . . . . 7 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ βˆƒπ‘₯ ∈ On 𝐴 βŠ† (β„΅β€˜π‘₯))
4645, 21sylib 217 . . . . . 6 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On)
4746, 24syl 17 . . . . 5 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On)
48 sbcan 3821 . . . . . 6 ([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦](𝑦 ∈ On ∧ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))) ↔ ([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦]𝑦 ∈ On ∧ [suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦](βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))))
49 sbcel1v 3840 . . . . . . . 8 ([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦]𝑦 ∈ On ↔ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On)
5049a1i 11 . . . . . . 7 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ ([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦]𝑦 ∈ On ↔ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On))
51 sbc3an 3839 . . . . . . . 8 ([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦](βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦)) ↔ ([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦]βˆ… ∈ 𝑦 ∧ [suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦]𝐴 βŠ† (β„΅β€˜π‘¦) ∧ [suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦](cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦)))
52 sbcel2gv 3841 . . . . . . . . 9 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ ([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦]βˆ… ∈ 𝑦 ↔ βˆ… ∈ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
53 sbcssg 4515 . . . . . . . . . 10 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ ([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦]𝐴 βŠ† (β„΅β€˜π‘¦) ↔ ⦋suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œπ΄ βŠ† ⦋suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œ(β„΅β€˜π‘¦)))
54 csbconstg 3904 . . . . . . . . . . 11 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ ⦋suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œπ΄ = 𝐴)
55 csbfv2g 6930 . . . . . . . . . . . 12 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ ⦋suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œ(β„΅β€˜π‘¦) = (β„΅β€˜β¦‹suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œπ‘¦))
56 csbvarg 4423 . . . . . . . . . . . . 13 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ ⦋suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œπ‘¦ = suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})
5756fveq2d 6885 . . . . . . . . . . . 12 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ (β„΅β€˜β¦‹suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œπ‘¦) = (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
5855, 57eqtrd 2764 . . . . . . . . . . 11 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ ⦋suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œ(β„΅β€˜π‘¦) = (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
5954, 58sseq12d 4007 . . . . . . . . . 10 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ (⦋suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œπ΄ βŠ† ⦋suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œ(β„΅β€˜π‘¦) ↔ 𝐴 βŠ† (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
6053, 59bitrd 279 . . . . . . . . 9 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ ([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦]𝐴 βŠ† (β„΅β€˜π‘¦) ↔ 𝐴 βŠ† (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
61 sbceqg 4401 . . . . . . . . . 10 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ ([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦](cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦) ↔ ⦋suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œ(cfβ€˜(β„΅β€˜π‘¦)) = ⦋suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œ(β„΅β€˜π‘¦)))
62 csbfv2g 6930 . . . . . . . . . . . 12 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ ⦋suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œ(cfβ€˜(β„΅β€˜π‘¦)) = (cfβ€˜β¦‹suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œ(β„΅β€˜π‘¦)))
6358fveq2d 6885 . . . . . . . . . . . 12 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ (cfβ€˜β¦‹suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œ(β„΅β€˜π‘¦)) = (cfβ€˜(β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
6462, 63eqtrd 2764 . . . . . . . . . . 11 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ ⦋suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œ(cfβ€˜(β„΅β€˜π‘¦)) = (cfβ€˜(β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
6564, 58eqeq12d 2740 . . . . . . . . . 10 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ (⦋suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œ(cfβ€˜(β„΅β€˜π‘¦)) = ⦋suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / π‘¦β¦Œ(β„΅β€˜π‘¦) ↔ (cfβ€˜(β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})) = (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
6661, 65bitrd 279 . . . . . . . . 9 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ ([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦](cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦) ↔ (cfβ€˜(β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})) = (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))
6752, 60, 663anbi123d 1432 . . . . . . . 8 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ (([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦]βˆ… ∈ 𝑦 ∧ [suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦]𝐴 βŠ† (β„΅β€˜π‘¦) ∧ [suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦](cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦)) ↔ (βˆ… ∈ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∧ 𝐴 βŠ† (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∧ (cfβ€˜(β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})) = (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))))
6851, 67bitrid 283 . . . . . . 7 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ ([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦](βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦)) ↔ (βˆ… ∈ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∧ 𝐴 βŠ† (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∧ (cfβ€˜(β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})) = (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))))
6950, 68anbi12d 630 . . . . . 6 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ (([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦]𝑦 ∈ On ∧ [suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦](βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))) ↔ (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On ∧ (βˆ… ∈ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∧ 𝐴 βŠ† (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∧ (cfβ€˜(β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})) = (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))))
7048, 69bitrid 283 . . . . 5 (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On β†’ ([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦](𝑦 ∈ On ∧ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))) ↔ (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On ∧ (βˆ… ∈ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∧ 𝐴 βŠ† (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∧ (cfβ€˜(β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})) = (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))))
7147, 70syl 17 . . . 4 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ ([suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦](𝑦 ∈ On ∧ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))) ↔ (suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∈ On ∧ (βˆ… ∈ suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} ∧ 𝐴 βŠ† (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}) ∧ (cfβ€˜(β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})) = (β„΅β€˜suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)})))))
7241, 71mpbird 257 . . 3 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ [suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} / 𝑦](𝑦 ∈ On ∧ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))))
7372spesbcd 3869 . 2 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ βˆƒπ‘¦(𝑦 ∈ On ∧ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))))
74 onintrab2 7778 . . 3 (βˆƒπ‘¦ ∈ On (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦)) ↔ ∩ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))} ∈ On)
75 df-rex 3063 . . 3 (βˆƒπ‘¦ ∈ On (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦)) ↔ βˆƒπ‘¦(𝑦 ∈ On ∧ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))))
76 risset 3222 . . 3 (∩ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))} ∈ On ↔ βˆƒπ‘₯ ∈ On π‘₯ = ∩ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))})
7774, 75, 763bitr3i 301 . 2 (βˆƒπ‘¦(𝑦 ∈ On ∧ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))) ↔ βˆƒπ‘₯ ∈ On π‘₯ = ∩ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))})
7873, 77sylib 217 1 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ βˆƒπ‘₯ ∈ On π‘₯ = ∩ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))})
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  βˆƒwrex 3062  {crab 3424  [wsbc 3769  β¦‹csb 3885   βˆ– cdif 3937   βŠ† wss 3940  βˆ…c0 4314  βˆ© cint 4940  ran crn 5667  Ord word 6353  Oncon0 6354  suc csuc 6356  β€˜cfv 6533  Ο‰com 7848  cardccrd 9925  β„΅cale 9926  cfccf 9927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9631  ax-ac2 10453
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8698  df-map 8817  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-oi 9500  df-har 9547  df-card 9929  df-aleph 9930  df-cf 9931  df-acn 9932  df-ac 10106
This theorem is referenced by:  minregex2  42741
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