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Theorem minregex 41813
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 3921 . . . . . . 7 (𝐴 ∈ (ran card βˆ– Ο‰) ↔ (𝐴 ∈ ran card ∧ Β¬ 𝐴 ∈ Ο‰))
2 omelon 9583 . . . . . . . . . 10 Ο‰ ∈ On
3 cardon 9881 . . . . . . . . . . 11 (cardβ€˜π΄) ∈ On
4 eleq1 2826 . . . . . . . . . . 11 ((cardβ€˜π΄) = 𝐴 β†’ ((cardβ€˜π΄) ∈ On ↔ 𝐴 ∈ On))
53, 4mpbii 232 . . . . . . . . . 10 ((cardβ€˜π΄) = 𝐴 β†’ 𝐴 ∈ On)
6 ontri1 6352 . . . . . . . . . 10 ((Ο‰ ∈ On ∧ 𝐴 ∈ On) β†’ (Ο‰ βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ Ο‰))
72, 5, 6sylancr 588 . . . . . . . . 9 ((cardβ€˜π΄) = 𝐴 β†’ (Ο‰ βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ Ο‰))
87pm5.32i 576 . . . . . . . 8 (((cardβ€˜π΄) = 𝐴 ∧ Ο‰ βŠ† 𝐴) ↔ ((cardβ€˜π΄) = 𝐴 ∧ Β¬ 𝐴 ∈ Ο‰))
9 iscard4 41812 . . . . . . . . 9 ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 ∈ ran card)
109anbi1i 625 . . . . . . . 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 462 . . . . . . 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 10027 . . . . . . . 8 (Ο‰ βŠ† 𝐴 β†’ ((cardβ€˜π΄) = 𝐴 ↔ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯)))
1615biimpa 478 . . . . . . 7 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ βˆƒπ‘₯ ∈ On 𝐴 = (β„΅β€˜π‘₯))
17 eqimss 4001 . . . . . . . . 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 3168 . . . . . . 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 7733 . . . . . 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 486 . . . . . . 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 7747 . . . . . . 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 6328 . . . . . . . . 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 7771 . . . . . . . 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 10026 . . . . . . . . 9 ((Ο‰ βŠ† 𝐴 ∧ (cardβ€˜π΄) = 𝐴) β†’ 𝐴 = (β„΅β€˜βˆ© {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}))
3130adantr 482 . . . . . . . 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 6398 . . . . . . . . 9 ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)} βŠ† suc ∩ {π‘₯ ∈ On ∣ 𝐴 βŠ† (β„΅β€˜π‘₯)}
33 alephord3 10015 . . . . . . . . . 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 586 . . . . . . . . 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 3983 . . . . . . 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 10519 . . . . . . . 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 1129 . . . . . 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 513 . . . . 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 586 . . . 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 3168 . . . . . . . 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 3792 . . . . . 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 3811 . . . . . . . 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 3810 . . . . . . . 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 3812 . . . . . . . . 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 4482 . . . . . . . . . 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 3875 . . . . . . . . . . 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 6892 . . . . . . . . . . . 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 4392 . . . . . . . . . . . . 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 6847 . . . . . . . . . . . 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 2777 . . . . . . . . . . 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 3978 . . . . . . . . . 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 4370 . . . . . . . . . 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 6892 . . . . . . . . . . . 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 6847 . . . . . . . . . . . 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 2777 . . . . . . . . . . 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 2753 . . . . . . . . . 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 1437 . . . . . . . 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 632 . . . . . 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 3840 . 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 7733 . . 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 3075 . . 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 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆƒwrex 3074  {crab 3408  [wsbc 3740  β¦‹csb 3856   βˆ– cdif 3908   βŠ† wss 3911  βˆ…c0 4283  βˆ© cint 4908  ran crn 5635  Ord word 6317  Oncon0 6318  suc csuc 6320  β€˜cfv 6497  Ο‰com 7803  cardccrd 9872  β„΅cale 9873  cfccf 9874
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9578  ax-ac2 10400
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8649  df-map 8768  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-oi 9447  df-har 9494  df-card 9876  df-aleph 9877  df-cf 9878  df-acn 9879  df-ac 10053
This theorem is referenced by:  minregex2  41814
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