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Theorem minregex 42270
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 3957 . . . . . . 7 (𝐴 ∈ (ran card βˆ– Ο‰) ↔ (𝐴 ∈ ran card ∧ Β¬ 𝐴 ∈ Ο‰))
2 omelon 9637 . . . . . . . . . 10 Ο‰ ∈ On
3 cardon 9935 . . . . . . . . . . 11 (cardβ€˜π΄) ∈ On
4 eleq1 2821 . . . . . . . . . . 11 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) ∈ On ↔ 𝐴 ∈ On))
53, 4mpbii 232 . . . . . . . . . 10 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
6 ontri1 6395 . . . . . . . . . 10 ((Ο‰ ∈ On ∧ 𝐴 ∈ On) β†’ (Ο‰ βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ Ο‰))
72, 5, 6sylancr 587 . . . . . . . . 9 ((cardβ€˜π΄) = 𝐴 β†’ (Ο‰ βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ Ο‰))
87pm5.32i 575 . . . . . . . 8 (((cardβ€˜π΄) = 𝐴 ∧ Ο‰ βŠ† 𝐴) ↔ ((cardβ€˜π΄) = 𝐴 ∧ Β¬ 𝐴 ∈ Ο‰))
9 iscard4 42269 . . . . . . . . 9 ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 ∈ ran card)
109anbi1i 624 . . . . . . . 8 (((cardβ€˜π΄) = 𝐴 ∧ Β¬ 𝐴 ∈ Ο‰) ↔ (𝐴 ∈ ran card ∧ Β¬ 𝐴 ∈ Ο‰))
118, 10bitr2i 275 . . . . . . 7 ((𝐴 ∈ ran card ∧ Β¬ 𝐴 ∈ Ο‰) ↔ ((cardβ€˜π΄) = 𝐴 ∧ Ο‰ βŠ† 𝐴))
12 ancom 461 . . . . . . 7 (((cardβ€˜π΄) = 𝐴 ∧ Ο‰ βŠ† 𝐴) ↔ (Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴))
131, 11, 123bitri 296 . . . . . 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 10081 . . . . . . . 8 (Ο‰ βŠ† 𝐴 β†’ ((cardβ€˜π΄) = 𝐴 ↔ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯)))
1615biimpa 477 . . . . . . 7 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯))
17 eqimss 4039 . . . . . . . . 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 3170 . . . . . . 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 7781 . . . . . 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 485 . . . . . . 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 7795 . . . . . . 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 6371 . . . . . . . . 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 7819 . . . . . . . 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 10080 . . . . . . . . 9 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ 𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
3130adantr 481 . . . . . . . 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 6441 . . . . . . . . 9 ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} βŠ† suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}
33 alephord3 10069 . . . . . . . . . 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 585 . . . . . . . . 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 4019 . . . . . . 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 10573 . . . . . . . 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 1128 . . . . . 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 512 . . . . 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 585 . . . 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 3170 . . . . . . . 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 3828 . . . . . 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 3847 . . . . . . . 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 3846 . . . . . . . 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 3848 . . . . . . . . 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 4522 . . . . . . . . . 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 3911 . . . . . . . . . . 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 6937 . . . . . . . . . . . 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 4430 . . . . . . . . . . . . 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 6892 . . . . . . . . . . . 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 2772 . . . . . . . . . . 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 4014 . . . . . . . . . 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 278 . . . . . . . . 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 4408 . . . . . . . . . 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 6937 . . . . . . . . . . . 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 6892 . . . . . . . . . . . 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 2772 . . . . . . . . . . 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 2748 . . . . . . . . . 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 278 . . . . . . . . 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 1436 . . . . . . . 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 282 . . . . . . 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 631 . . . . . 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 282 . . . . 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 256 . . 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 3876 . 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 7781 . . 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 3071 . . 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 3230 . . 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 300 . 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 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  βˆƒwrex 3070  {crab 3432  [wsbc 3776  β¦‹csb 3892   βˆ– cdif 3944   βŠ† wss 3947  βˆ…c0 4321  βˆ© cint 4949  ran crn 5676  Ord word 6360  Oncon0 6361  suc csuc 6363  β€˜cfv 6540  Ο‰com 7851  cardccrd 9926  β„΅cale 9927  cfccf 9928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-ac2 10454
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-oi 9501  df-har 9548  df-card 9930  df-aleph 9931  df-cf 9932  df-acn 9933  df-ac 10107
This theorem is referenced by:  minregex2  42271
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