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Mirrors > Home > MPE Home > Th. List > Mathboxes > nobndlem4 | Structured version Visualization version GIF version |
Description: Lemma for nobndup 31099 and nobnddown 31100. The infimum of the class of all ordinals such that 𝐴 is not 𝑋 is an ordinal. (Contributed by Scott Fenton, 17-Aug-2011.) |
Ref | Expression |
---|---|
nobndlem4.1 | ⊢ 𝑋 ∈ {1𝑜, 2𝑜} |
Ref | Expression |
---|---|
nobndlem4 | ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdayelon 31079 | . . 3 ⊢ ( bday ‘𝐴) ∈ On | |
2 | nobndlem4.1 | . . . . 5 ⊢ 𝑋 ∈ {1𝑜, 2𝑜} | |
3 | 2 | nosgnn0i 31056 | . . . 4 ⊢ ∅ ≠ 𝑋 |
4 | fvnobday 31081 | . . . . 5 ⊢ (𝐴 ∈ No → (𝐴‘( bday ‘𝐴)) = ∅) | |
5 | 4 | neeq1d 2841 | . . . 4 ⊢ (𝐴 ∈ No → ((𝐴‘( bday ‘𝐴)) ≠ 𝑋 ↔ ∅ ≠ 𝑋)) |
6 | 3, 5 | mpbiri 247 | . . 3 ⊢ (𝐴 ∈ No → (𝐴‘( bday ‘𝐴)) ≠ 𝑋) |
7 | fveq2 6103 | . . . . 5 ⊢ (𝑥 = ( bday ‘𝐴) → (𝐴‘𝑥) = (𝐴‘( bday ‘𝐴))) | |
8 | 7 | neeq1d 2841 | . . . 4 ⊢ (𝑥 = ( bday ‘𝐴) → ((𝐴‘𝑥) ≠ 𝑋 ↔ (𝐴‘( bday ‘𝐴)) ≠ 𝑋)) |
9 | 8 | rspcev 3282 | . . 3 ⊢ ((( bday ‘𝐴) ∈ On ∧ (𝐴‘( bday ‘𝐴)) ≠ 𝑋) → ∃𝑥 ∈ On (𝐴‘𝑥) ≠ 𝑋) |
10 | 1, 6, 9 | sylancr 694 | . 2 ⊢ (𝐴 ∈ No → ∃𝑥 ∈ On (𝐴‘𝑥) ≠ 𝑋) |
11 | onintrab2 6894 | . 2 ⊢ (∃𝑥 ∈ On (𝐴‘𝑥) ≠ 𝑋 ↔ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ On) | |
12 | 10, 11 | sylib 207 | 1 ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ 𝑋} ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 {crab 2900 ∅c0 3874 {cpr 4127 ∩ cint 4410 Oncon0 5640 ‘cfv 5804 1𝑜c1o 7440 2𝑜c2o 7441 No csur 31037 bday cbday 31039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-1o 7447 df-2o 7448 df-no 31040 df-bday 31042 |
This theorem is referenced by: nobndup 31099 nobnddown 31100 |
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