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Mirrors > Home > MPE Home > Th. List > infeq5i | Structured version Visualization version GIF version |
Description: Half of infeq5 9094. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
infeq5i | ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difexg 5223 | . 2 ⊢ (ω ∈ V → (ω ∖ {∅}) ∈ V) | |
2 | 0ex 5203 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 2 | snid 4594 | . . . 4 ⊢ ∅ ∈ {∅} |
4 | disj4 4407 | . . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ¬ (ω ∖ {∅}) ⊊ ω) | |
5 | disj3 4402 | . . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ω = (ω ∖ {∅})) | |
6 | 4, 5 | bitr3i 279 | . . . . 5 ⊢ (¬ (ω ∖ {∅}) ⊊ ω ↔ ω = (ω ∖ {∅})) |
7 | peano1 7595 | . . . . . . 7 ⊢ ∅ ∈ ω | |
8 | eleq2 2901 | . . . . . . 7 ⊢ (ω = (ω ∖ {∅}) → (∅ ∈ ω ↔ ∅ ∈ (ω ∖ {∅}))) | |
9 | 7, 8 | mpbii 235 | . . . . . 6 ⊢ (ω = (ω ∖ {∅}) → ∅ ∈ (ω ∖ {∅})) |
10 | 9 | eldifbd 3948 | . . . . 5 ⊢ (ω = (ω ∖ {∅}) → ¬ ∅ ∈ {∅}) |
11 | 6, 10 | sylbi 219 | . . . 4 ⊢ (¬ (ω ∖ {∅}) ⊊ ω → ¬ ∅ ∈ {∅}) |
12 | 3, 11 | mt4 116 | . . 3 ⊢ (ω ∖ {∅}) ⊊ ω |
13 | unidif0 5252 | . . . . 5 ⊢ ∪ (ω ∖ {∅}) = ∪ ω | |
14 | limom 7589 | . . . . . 6 ⊢ Lim ω | |
15 | limuni 6245 | . . . . . 6 ⊢ (Lim ω → ω = ∪ ω) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ ω = ∪ ω |
17 | 13, 16 | eqtr4i 2847 | . . . 4 ⊢ ∪ (ω ∖ {∅}) = ω |
18 | 17 | psseq2i 4066 | . . 3 ⊢ ((ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) ↔ (ω ∖ {∅}) ⊊ ω) |
19 | 12, 18 | mpbir 233 | . 2 ⊢ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) |
20 | psseq1 4063 | . . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ 𝑥)) | |
21 | unieq 4839 | . . . . 5 ⊢ (𝑥 = (ω ∖ {∅}) → ∪ 𝑥 = ∪ (ω ∖ {∅})) | |
22 | 21 | psseq2d 4069 | . . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → ((ω ∖ {∅}) ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}))) |
23 | 20, 22 | bitrd 281 | . . 3 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}))) |
24 | 23 | spcegv 3596 | . 2 ⊢ ((ω ∖ {∅}) ∈ V → ((ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) → ∃𝑥 𝑥 ⊊ ∪ 𝑥)) |
25 | 1, 19, 24 | mpisyl 21 | 1 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∃wex 1776 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 ∩ cin 3934 ⊊ wpss 3936 ∅c0 4290 {csn 4560 ∪ cuni 4831 Lim wlim 6186 ωcom 7574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-tr 5165 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-om 7575 |
This theorem is referenced by: infeq5 9094 inf5 9102 |
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