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Theorem cfeq0 10251
Description: Only the ordinal zero has cofinality zero. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)
Assertion
Ref Expression
cfeq0 (𝐴 ∈ On β†’ ((cfβ€˜π΄) = βˆ… ↔ 𝐴 = βˆ…))

Proof of Theorem cfeq0
Dummy variables 𝑣 𝑀 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfval 10242 . . . 4 (𝐴 ∈ On β†’ (cfβ€˜π΄) = ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
21eqeq1d 2735 . . 3 (𝐴 ∈ On β†’ ((cfβ€˜π΄) = βˆ… ↔ ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} = βˆ…))
3 vex 3479 . . . . . . . . 9 𝑣 ∈ V
4 eqeq1 2737 . . . . . . . . . . 11 (π‘₯ = 𝑣 β†’ (π‘₯ = (cardβ€˜π‘¦) ↔ 𝑣 = (cardβ€˜π‘¦)))
54anbi1d 631 . . . . . . . . . 10 (π‘₯ = 𝑣 β†’ ((π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ (𝑣 = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
65exbidv 1925 . . . . . . . . 9 (π‘₯ = 𝑣 β†’ (βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ βˆƒπ‘¦(𝑣 = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
73, 6elab 3669 . . . . . . . 8 (𝑣 ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} ↔ βˆƒπ‘¦(𝑣 = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
8 fveq2 6892 . . . . . . . . . . . 12 (𝑣 = (cardβ€˜π‘¦) β†’ (cardβ€˜π‘£) = (cardβ€˜(cardβ€˜π‘¦)))
9 cardidm 9954 . . . . . . . . . . . 12 (cardβ€˜(cardβ€˜π‘¦)) = (cardβ€˜π‘¦)
108, 9eqtrdi 2789 . . . . . . . . . . 11 (𝑣 = (cardβ€˜π‘¦) β†’ (cardβ€˜π‘£) = (cardβ€˜π‘¦))
11 eqeq2 2745 . . . . . . . . . . 11 (𝑣 = (cardβ€˜π‘¦) β†’ ((cardβ€˜π‘£) = 𝑣 ↔ (cardβ€˜π‘£) = (cardβ€˜π‘¦)))
1210, 11mpbird 257 . . . . . . . . . 10 (𝑣 = (cardβ€˜π‘¦) β†’ (cardβ€˜π‘£) = 𝑣)
1312adantr 482 . . . . . . . . 9 ((𝑣 = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ (cardβ€˜π‘£) = 𝑣)
1413exlimiv 1934 . . . . . . . 8 (βˆƒπ‘¦(𝑣 = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ (cardβ€˜π‘£) = 𝑣)
157, 14sylbi 216 . . . . . . 7 (𝑣 ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} β†’ (cardβ€˜π‘£) = 𝑣)
16 cardon 9939 . . . . . . 7 (cardβ€˜π‘£) ∈ On
1715, 16eqeltrrdi 2843 . . . . . 6 (𝑣 ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} β†’ 𝑣 ∈ On)
1817ssriv 3987 . . . . 5 {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} βŠ† On
19 onint0 7779 . . . . 5 ({π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} βŠ† On β†’ (∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} = βˆ… ↔ βˆ… ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))}))
2018, 19ax-mp 5 . . . 4 (∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} = βˆ… ↔ βˆ… ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
21 0ex 5308 . . . . . 6 βˆ… ∈ V
22 eqeq1 2737 . . . . . . . 8 (π‘₯ = βˆ… β†’ (π‘₯ = (cardβ€˜π‘¦) ↔ βˆ… = (cardβ€˜π‘¦)))
2322anbi1d 631 . . . . . . 7 (π‘₯ = βˆ… β†’ ((π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ (βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
2423exbidv 1925 . . . . . 6 (π‘₯ = βˆ… β†’ (βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ βˆƒπ‘¦(βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
2521, 24elab 3669 . . . . 5 (βˆ… ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} ↔ βˆƒπ‘¦(βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
26 onss 7772 . . . . . . . . . . 11 (𝐴 ∈ On β†’ 𝐴 βŠ† On)
27 sstr 3991 . . . . . . . . . . . 12 ((𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† On) β†’ 𝑦 βŠ† On)
2827ancoms 460 . . . . . . . . . . 11 ((𝐴 βŠ† On ∧ 𝑦 βŠ† 𝐴) β†’ 𝑦 βŠ† On)
2926, 28sylan 581 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 βŠ† 𝐴) β†’ 𝑦 βŠ† On)
30293adant2 1132 . . . . . . . . 9 ((𝐴 ∈ On ∧ βˆ… = (cardβ€˜π‘¦) ∧ 𝑦 βŠ† 𝐴) β†’ 𝑦 βŠ† On)
31303adant3r 1182 . . . . . . . 8 ((𝐴 ∈ On ∧ βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ 𝑦 βŠ† On)
32 simp2 1138 . . . . . . . 8 ((𝐴 ∈ On ∧ βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ βˆ… = (cardβ€˜π‘¦))
33 simp3 1139 . . . . . . . 8 ((𝐴 ∈ On ∧ βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))
34 eqcom 2740 . . . . . . . . . . . 12 (βˆ… = (cardβ€˜π‘¦) ↔ (cardβ€˜π‘¦) = βˆ…)
35 vex 3479 . . . . . . . . . . . . . 14 𝑦 ∈ V
36 onssnum 10035 . . . . . . . . . . . . . 14 ((𝑦 ∈ V ∧ 𝑦 βŠ† On) β†’ 𝑦 ∈ dom card)
3735, 36mpan 689 . . . . . . . . . . . . 13 (𝑦 βŠ† On β†’ 𝑦 ∈ dom card)
38 cardnueq0 9959 . . . . . . . . . . . . 13 (𝑦 ∈ dom card β†’ ((cardβ€˜π‘¦) = βˆ… ↔ 𝑦 = βˆ…))
3937, 38syl 17 . . . . . . . . . . . 12 (𝑦 βŠ† On β†’ ((cardβ€˜π‘¦) = βˆ… ↔ 𝑦 = βˆ…))
4034, 39bitrid 283 . . . . . . . . . . 11 (𝑦 βŠ† On β†’ (βˆ… = (cardβ€˜π‘¦) ↔ 𝑦 = βˆ…))
4140biimpa 478 . . . . . . . . . 10 ((𝑦 βŠ† On ∧ βˆ… = (cardβ€˜π‘¦)) β†’ 𝑦 = βˆ…)
42 sseq1 4008 . . . . . . . . . . . 12 (𝑦 = βˆ… β†’ (𝑦 βŠ† 𝐴 ↔ βˆ… βŠ† 𝐴))
43 rexeq 3322 . . . . . . . . . . . . 13 (𝑦 = βˆ… β†’ (βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀 ↔ βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀))
4443ralbidv 3178 . . . . . . . . . . . 12 (𝑦 = βˆ… β†’ (βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀 ↔ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀))
4542, 44anbi12d 632 . . . . . . . . . . 11 (𝑦 = βˆ… β†’ ((𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀) ↔ (βˆ… βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀)))
4645biimpa 478 . . . . . . . . . 10 ((𝑦 = βˆ… ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ (βˆ… βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀))
4741, 46sylan 581 . . . . . . . . 9 (((𝑦 βŠ† On ∧ βˆ… = (cardβ€˜π‘¦)) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ (βˆ… βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀))
48 rex0 4358 . . . . . . . . . . . . 13 Β¬ βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀
4948rgenw 3066 . . . . . . . . . . . 12 βˆ€π‘§ ∈ 𝐴 Β¬ βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀
50 r19.2z 4495 . . . . . . . . . . . 12 ((𝐴 β‰  βˆ… ∧ βˆ€π‘§ ∈ 𝐴 Β¬ βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀) β†’ βˆƒπ‘§ ∈ 𝐴 Β¬ βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀)
5149, 50mpan2 690 . . . . . . . . . . 11 (𝐴 β‰  βˆ… β†’ βˆƒπ‘§ ∈ 𝐴 Β¬ βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀)
52 rexnal 3101 . . . . . . . . . . 11 (βˆƒπ‘§ ∈ 𝐴 Β¬ βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀 ↔ Β¬ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀)
5351, 52sylib 217 . . . . . . . . . 10 (𝐴 β‰  βˆ… β†’ Β¬ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀)
5453necon4ai 2973 . . . . . . . . 9 (βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ βˆ… 𝑧 βŠ† 𝑀 β†’ 𝐴 = βˆ…)
5547, 54simpl2im 505 . . . . . . . 8 (((𝑦 βŠ† On ∧ βˆ… = (cardβ€˜π‘¦)) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ 𝐴 = βˆ…)
5631, 32, 33, 55syl21anc 837 . . . . . . 7 ((𝐴 ∈ On ∧ βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ 𝐴 = βˆ…)
57563expib 1123 . . . . . 6 (𝐴 ∈ On β†’ ((βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ 𝐴 = βˆ…))
5857exlimdv 1937 . . . . 5 (𝐴 ∈ On β†’ (βˆƒπ‘¦(βˆ… = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) β†’ 𝐴 = βˆ…))
5925, 58biimtrid 241 . . . 4 (𝐴 ∈ On β†’ (βˆ… ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} β†’ 𝐴 = βˆ…))
6020, 59biimtrid 241 . . 3 (𝐴 ∈ On β†’ (∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} = βˆ… β†’ 𝐴 = βˆ…))
612, 60sylbid 239 . 2 (𝐴 ∈ On β†’ ((cfβ€˜π΄) = βˆ… β†’ 𝐴 = βˆ…))
62 fveq2 6892 . . 3 (𝐴 = βˆ… β†’ (cfβ€˜π΄) = (cfβ€˜βˆ…))
63 cf0 10246 . . 3 (cfβ€˜βˆ…) = βˆ…
6462, 63eqtrdi 2789 . 2 (𝐴 = βˆ… β†’ (cfβ€˜π΄) = βˆ…)
6561, 64impbid1 224 1 (𝐴 ∈ On β†’ ((cfβ€˜π΄) = βˆ… ↔ 𝐴 = βˆ…))
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  {cab 2710   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βŠ† wss 3949  βˆ…c0 4323  βˆ© cint 4951  dom cdm 5677  Oncon0 6365  β€˜cfv 6544  cardccrd 9930  cfccf 9932
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-er 8703  df-en 8940  df-dom 8941  df-card 9934  df-cf 9936
This theorem is referenced by: (None)
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